Showing that an eigenbasis makes for good coordinate systems

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:03.870
I've talked a lot about the idea
that eigenvectors could

00:00:03.870 --> 00:00:07.650
make for good bases or
good basis vectors.

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So let's explore that idea
a little bit more.

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Let's say I have some
transformation.

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Let's say it's a transformation
from Rn to Rn,

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and it can be represented
by the matrix, A.

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So the transformation of x
is equal to the n-by-n

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matrix, A times x.

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Now let's say that we have
n linearly independent

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eigenvectors of A.

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And this isn't always going to
be the case, but it can often

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be the case.

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It's definitely possible.

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Let's assume that A has
n linearly independent

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eigenvectors.

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So I'm going to call them v1,
v2, all the way through vn.

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Now, n linearly independent
vectors in Rn can definitely

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be a basis for Rn.

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We've seen that multiple
times.

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And what I want to show you in
this video is that this makes

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a particularly good basis
for this transformation.

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So let's explore that.

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So the transformation of each of
these vectors-- I'll write

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it over here.

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The transformation of vector 1
is equal to A times vector 1

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and since vector 1 is an
eigenvector of A, that's going

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to be equal to some eigenvalue
lambda 1 times vector 1.

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We could do that for
all of them.

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The transformation of vector 2
is equal to A times v2, which

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is equal to some eigenvalue
lambda 2 times v2.

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And I'm just going to skip all
of them and just go straight

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to the nth one.

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We have n of these
eigenvectors.

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You might have a lot more.

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We're just assuming that A
has at least n linearly

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independent eigenvectors.

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In general, you could take
scaled up versions of these

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and they'll also be
eigenvectors.

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Let's see, so the transformation
of vn is going

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to be equal to A times vn.

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And because these are all
eigenvectors, A times vn is

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just going to be lambda n,
some eigenvalue times the

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vector, vn.

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Now, what are these
also equal to?

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Well, this is equal to, and this
is probably going to be

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unbelievably obvious to you, but
this is the same thing as

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lambda 1 times vn plus 0
times v2 plus all the

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way to 0 times vn.

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And this right here is going to
be 0 times v1 plus lambda 2

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times v2 plus all the way,
0 times all of the

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other vectors vn.

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And then this guy down here,
this is going to be 0 times v1

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plus 0 times v2 plus 0 times
all of these basis vectors,

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these eigenvectors, but
lambda n times vn.

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This is almost stunningly
obvious, right?

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I just rewrote this as this plus
a bunch of zero vectors.

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But the reason why I wrote that
is, because in a second,

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we're going to take this as a
basis and we're going to find

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coordinates with respect to that
basis, and so this guy's

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coordinates will be lambda 1,
0, 0, because that's the

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coefficients on our
basis vectors.

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So let's do that.

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So let's say that we define
this as some basis.

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So B is equal to the set of--
actually, I don't even have to

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write it that way.

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Let's say I say that B,
I have some basis B,

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that's equal to that.

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What I want to show you is that
when I do a change of

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basis-- we've seen this before--
in my standard

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coordinates or in coordinates
with respect to the standard

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basis, you give me some vector
in Rn, I'm going to multiply

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it times A, and you're
going to have the

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transformation of it.

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It's also going to be in Rn.

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Now, we know we can do
a change of basis.

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And in a change of basis, if you
want to go that way, you

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multiply by C inverse, which
is-- remember, the change of

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basis matrix C, if you want to
go in this direction, you

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multiply by C.

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The change of basis matrix is
just a matrix with all of

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these vectors as columns.

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It's very easy to construct.

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But if you change your basis
from x to our new basis, you

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multiply it by the
inverse of that.

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We've seen that multiple
times.

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If they're all orthonormal, then
this is the same thing as

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a transpose.

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We can't assume that, though.

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And so this is going to
be x in our new basis.

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And if we want to find some
transformation, if we want to

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find the transformation matrix
for T with respect to our new

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basis, it's going to
be some matrix D.

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And if you multiply D times x,
you're going to get this guy,

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but you're going to get the B
representation of that guy.

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The transformation of the vector
x is B representation.

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And if we want to go back and
forth between that guy and

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that guy, if we want to go in
this direction, you can

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multiply this times C, and
you'll just get the

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transformation of x.

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And if you want to go in that
direction, you could multiply

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by the inverse of your change
of basis matrix.

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We've seen this multiple
times already.

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But what I've claimed or I've
kind of hinted at is that if I

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have a basis that's defined by
eigenvectors of A, that this

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will be a very nice matrix,
that this might be the

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coordinate system that you want
to operate in, especially

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if you're going to apply
this matrix a lot.

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If you're going to do this
transformation on a lot of

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different things, you're going
to do it over and over and

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over again, maybe to the same
set, then it maybe is worth

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the overhead to do the
conversion and just use this

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as your coordinate system.

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So let's see that this will
actually be a nice-looking,

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easy-to-compute-with and
actually diagonal matrix.

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So we know that the
transformation-- what is the

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transformation of-- let's
write this in a bunch of

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different formats.

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Let me scroll down
a little bit.

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So if I wanted to write the
transformation of v1 in B

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coordinates, what would it be?

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It's just going to be equal to--
well, these are the basis

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vectors, right?

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So it's the coefficient
on the basis vectors.

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So it's going to be equal to
lambda 1, and then there's a

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bunch of zeroes.

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It's lambda 1 times v1 plus 0
times v2 plus 0 times v3, all

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the way to 0 times vn.

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That's what it's equal to.

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But it's also equal to D, and
we can write D like this.

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D is also a transformation
between Rn and Rn, just a

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different coordinate system.

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So D is going to just be a bunch
of column vectors d1,

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d2, all the way through dn
times-- this is the same thing

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as D times our B representation

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of the vector v1.

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But what is our B
representation

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of the vector v1?

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Well, the vector, v1 is just 1
times v1 plus 0 times v2 plus

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0 times v3 all the way
to 0 times vn.

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v1 is a basis vector.

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That's just 1 times itself plus
0 times everything else.

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So this is what its
representation is in the B

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coordinate system.

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Now, what is this going
to be equal to?

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And we've seen this before.

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This is all a bit of review.

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I might even be boring you.

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This is just equal to 1 times
d1 plus 0 times d2 plus 0

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times all the other columns.

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This is just equal to d1.

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So just like that, we have our
first column of our matrix D.

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We could just keep doing that.

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I'll do it multiple times.

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The transformation of v2 in our
new coordinate system with

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respect to our new basis is
going to be equal to-- well,

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we know what the transformation
of v2 is.

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It's 0 times v1 plus lambda 2
times v2 and then plus 0 times

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everything else.

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And that's the same thing as D,
d1, d2, all the way through

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dn times our B representation
of vector 2.

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Well, vector 2 is one of
the basis vectors.

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It's just 0 times v1 plus 1
times v2 plus 0 times v3 all

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the way, the rest is 0.

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So what's this going
to be equal to?

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This is 0 times d1 plus
1 times d2 and 0 times

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everything else, so
it's equal to d2.

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I think you get the
general idea.

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I'll do it one more time
just to really

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hammer the point home.

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The transformation of the nth
basis vector, which is also an

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eigenvector of our original
matrix A or of our

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transformation in standard
coordinates, in B coordinates,

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is going to be equal to what?

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Well, we wrote it
right up here.

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It's going to be a
bunch of zeroes.

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It's 0 times all of these guys
plus lambda n times vn.

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And this is going to be this guy
d1, d2, all the way to dn

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times the B representation of
the nth basis vector, which is

00:09:24.930 --> 00:09:28.740
just 0 times v1, 0 times v2
and 0 times all of them,

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except for 1 times vn.

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And so this is going to be equal
to 0 times d1 plus 0

00:09:36.790 --> 00:09:39.210
times d2 plus 0 times all
of these guys all the

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way to 1 times dn.

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So that's going to
be equal to dn.

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And just like that, we know what
our transformation matrix

00:09:47.810 --> 00:09:50.310
is going to look like with
respect to this new basis,

00:09:50.310 --> 00:09:54.880
where this basis was defined or
it's made up of n linearly

00:09:54.880 --> 00:09:58.730
independent eigenvectors of
our original matrix A.

00:09:58.730 --> 00:10:01.560
So what does D look like?

00:10:01.560 --> 00:10:05.700
Our matrix D is going to look
like-- its first column is

00:10:05.700 --> 00:10:07.170
right there.

00:10:07.170 --> 00:10:08.490
We figured that one out.

00:10:08.490 --> 00:10:12.440
Lambda 1, and then we just
have a bunch of zeroes.

00:10:12.440 --> 00:10:14.640
Its second column
is right here.

00:10:14.640 --> 00:10:15.920
d2 is equal to this.

00:10:15.920 --> 00:10:20.040
It's 0, lambda 2, and then
a bunch of zeroes.

00:10:20.040 --> 00:10:21.610
And then this is in
general the case.

00:10:21.610 --> 00:10:24.010
The nth column is going to have
a zero everywhere except

00:10:24.010 --> 00:10:24.700
along the diagonal.

00:10:24.700 --> 00:10:25.985
It's going to be lambda n.

00:10:25.985 --> 00:10:28.620
It's going to be the eigenvalue
for the nth

00:10:28.620 --> 00:10:30.210
eigenvector.

00:10:30.210 --> 00:10:32.180
And so the diagonal is going to
look-- you're going to have

00:10:32.180 --> 00:10:35.510
lambda 3 all the way
down to lambda n.

00:10:35.510 --> 00:10:39.890
And our nth column is lambda n
with just a bunch of zeroes

00:10:39.890 --> 00:10:41.560
everywhere.

00:10:41.560 --> 00:10:46.130
So D, when we picked-- this
is a neat result.

00:10:46.130 --> 00:10:51.990
If A has n linearly independent
eigenvectors, and

00:10:51.990 --> 00:10:54.500
this isn't always the case,
but we can figure out that

00:10:54.500 --> 00:10:57.380
eigenvectors and say, hey, I can
take a collection of n of

00:10:57.380 --> 00:10:59.970
these that are linearly
independent, then those will

00:10:59.970 --> 00:11:02.350
be a basis for Rn.

00:11:02.350 --> 00:11:07.930
n linearly independent vectors
in Rn are a basis for Rn.

00:11:07.930 --> 00:11:12.080
But when you use that basis,
when you use the linearly

00:11:12.080 --> 00:11:15.320
independent eigenvectors of A
as a basis, we call this an

00:11:15.320 --> 00:11:16.570
eigenbasis.

00:11:21.450 --> 00:11:26.310
The transformation matrix with
respect to that eigenbasis, it

00:11:26.310 --> 00:11:29.220
becomes a very, very
nice matrix.

00:11:29.220 --> 00:11:31.230
This is super easy
to multiply.

00:11:31.230 --> 00:11:33.050
It's super easy to invert.

00:11:33.050 --> 00:11:35.370
It's super easy to take
the determinant of.

00:11:35.370 --> 00:11:36.330
We've seen it multiple times.

00:11:36.330 --> 00:11:37.900
It just has a ton of
neat properties.

00:11:37.900 --> 00:11:40.820
It's just a good basis
to be dealing with.

00:11:40.820 --> 00:11:43.030
So that's kind of the
big takeaway.

00:11:43.030 --> 00:11:45.780
In all of linear algebra, we did
all this stuff with spaces

00:11:45.780 --> 00:11:49.670
and vectors and all of that,
but in general, vectors are

00:11:49.670 --> 00:11:55.100
abstract representations
of real world things.

00:11:55.100 --> 00:11:58.010
You could represent a vector
as the stock returns or it

00:11:58.010 --> 00:12:00.540
could be a vector of weather
in a certain part of the

00:12:00.540 --> 00:12:02.730
country, and you can create
these spaces based on the

00:12:02.730 --> 00:12:04.440
number of dimensions
and all of that.

00:12:04.440 --> 00:12:06.090
And then you're going to
have transformations.

00:12:06.090 --> 00:12:08.690
Sometimes, like when we learn
about Markov chains, your

00:12:08.690 --> 00:12:10.780
transformations are essentially
what's the

00:12:10.780 --> 00:12:15.910
probability after one time
increment that something state

00:12:15.910 --> 00:12:17.980
will change to something else,
then you'll want to apply that

00:12:17.980 --> 00:12:22.230
matrix many, many, many, many
times to see what the stable

00:12:22.230 --> 00:12:23.520
state is for a lot of things.

00:12:23.520 --> 00:12:26.660
And I know I'm not explaining
any of this to you well, but I

00:12:26.660 --> 00:12:30.370
wanted to tell you that all of
linear algebra is really just

00:12:30.370 --> 00:12:34.140
a very general way to solve a
whole universe of problems.

00:12:34.140 --> 00:12:36.730
And what's useful about this is
you can have transformation

00:12:36.730 --> 00:12:39.390
matrices that define
these functions

00:12:39.390 --> 00:12:41.680
essentially on data sets.

00:12:41.680 --> 00:12:43.260
And what we've learned now is
that when you look at the

00:12:43.260 --> 00:12:46.130
eigenvectors and the
eigenvalues, you can change

00:12:46.130 --> 00:12:49.640
your bases so that you can solve
your problems in much

00:12:49.640 --> 00:12:50.360
simpler ways.

00:12:50.360 --> 00:12:52.950
And I know it's all very
abstract right now, but you

00:12:52.950 --> 00:12:55.560
now have the toolkit, and the
rest of your life, you have to

00:12:55.560 --> 00:12:59.300
figure out how to apply this
toolkit to specific problems

00:12:59.300 --> 00:13:04.510
in probability or statistics or
finance or modeling weather

00:13:04.510 --> 00:13:06.960
systems or who knows
what else.


Eigenvectors and eigenspaces for a 3x3 matrix

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.410
In the last video we set out to
find the eigenvalues values

00:00:03.410 --> 00:00:05.830
of this 3 by 3 matrix, A.

00:00:05.830 --> 00:00:08.280
And we said, look an eigenvalue
is any value,

00:00:08.280 --> 00:00:11.410
lambda, that satisfies
this equation if v

00:00:11.410 --> 00:00:13.280
is a non-zero vector.

00:00:13.280 --> 00:00:17.300
And that says, any value,
lambda, that satisfies this

00:00:17.300 --> 00:00:20.390
equation for v is a
non-zero vector.

00:00:20.390 --> 00:00:22.760
Then we just did a little bit
of I guess we could call it

00:00:22.760 --> 00:00:25.370
vector algebra up here
to come up with that.

00:00:25.370 --> 00:00:26.890
You can review that
video if you like.

00:00:26.890 --> 00:00:29.700
And then we determined, look
the only way that this is

00:00:29.700 --> 00:00:33.710
going to have a non-zero
solution is if this matrix has

00:00:33.710 --> 00:00:36.320
a non-trivial null space.

00:00:36.320 --> 00:00:39.790
And only non-invertible matrices
have a non-trivial

00:00:39.790 --> 00:00:40.970
null space.

00:00:40.970 --> 00:00:45.020
Or, only matrices that have
a determinant of 0 have

00:00:45.020 --> 00:00:46.770
non-trivial null spaces.

00:00:46.770 --> 00:00:49.510
So you do that, you got your
characteristic polynomial, and

00:00:49.510 --> 00:00:50.680
we were able to solve it.

00:00:50.680 --> 00:00:55.000
And we got our eigenvalues where
lambda is equal to 3 and

00:00:55.000 --> 00:00:58.370
lambda is equal to minus 3.

00:00:58.370 --> 00:01:01.050
So now, let's do-- what I
consider the more interesting

00:01:01.050 --> 00:01:03.840
part-- is actually find out
the eigenvectors or the

00:01:03.840 --> 00:01:05.530
eigenspaces.

00:01:05.530 --> 00:01:08.800
So we can go back to this
equation, for any eigenvalue

00:01:08.800 --> 00:01:09.570
this must be true.

00:01:09.570 --> 00:01:12.300
This must be true but this
is easier to work with.

00:01:12.300 --> 00:01:18.140
And so, this matrix right here
times your eigenvector must be

00:01:18.140 --> 00:01:21.100
equal 0 for any given
eigenvalue.

00:01:21.100 --> 00:01:23.720
This matrix right here--
I've just copied

00:01:23.720 --> 00:01:24.740
and pasted from above.

00:01:24.740 --> 00:01:27.120
I marked it up with the Rule
of Sarrus so you can ignore

00:01:27.120 --> 00:01:28.870
those lines-- is just
this matrix right

00:01:28.870 --> 00:01:30.430
here for any lambda.

00:01:30.430 --> 00:01:32.810
Lambda times the identity
matrix minus A

00:01:32.810 --> 00:01:34.330
ends up being this.

00:01:34.330 --> 00:01:37.600
So let's take this matrix for
each of our lambdas and then

00:01:37.600 --> 00:01:42.190
solve for our eigenvectors
or our eigenspaces.

00:01:42.190 --> 00:01:47.390
So let me take the case of
lambda is equal to 3 first. So

00:01:47.390 --> 00:01:52.290
if lambda is equal to 3, this
matrix becomes lambda plus 1

00:01:52.290 --> 00:01:58.900
is 4, lambda minus 2 is 1,
lambda minus 2 is 1.

00:01:58.900 --> 00:02:02.590
And then all of the other terms
stay the same, minus 2,

00:02:02.590 --> 00:02:08.380
minus 2, minus 2, 1,
minus 2 and 1.

00:02:08.380 --> 00:02:12.390
And then this times that vector,
v, or our eigenvector

00:02:12.390 --> 00:02:15.010
v is equal to 0.

00:02:15.010 --> 00:02:19.070
Or we could say that the
eigenspace for the eigenvalue

00:02:19.070 --> 00:02:21.990
3 is the null space
of this matrix.

00:02:21.990 --> 00:02:23.230
Which is not this matrix.

00:02:23.230 --> 00:02:25.690
It's lambda times the
identity minus A.

00:02:25.690 --> 00:02:29.060
So the null space of this matrix
is the eigenspace.

00:02:29.060 --> 00:02:32.510
So all of the values that
satisfy this make up the

00:02:32.510 --> 00:02:36.540
eigenvectors of the eigenspace
of lambda is equal to 3.

00:02:36.540 --> 00:02:37.400
So let's just solve for this.

00:02:37.400 --> 00:02:39.940
So the null space of this guy--
we could just put in

00:02:39.940 --> 00:02:42.750
reduced row echelon form-- the
null space of this guy is the

00:02:42.750 --> 00:02:44.580
same thing as the null space
of this guy in reduced row

00:02:44.580 --> 00:02:45.560
echelon form.

00:02:45.560 --> 00:02:48.310
So let's put this in reduced
row echelon form.

00:02:48.310 --> 00:02:51.650
So the first thing I
want to do-- let me

00:02:51.650 --> 00:02:54.070
just do it down here.

00:02:54.070 --> 00:02:58.940
So let me-- I'll keep my first
row the same for now.

00:02:58.940 --> 00:03:02.200
4 minus 2, minus 2.

00:03:02.200 --> 00:03:07.250
And let me replace my second row
with my second row times 2

00:03:07.250 --> 00:03:08.150
plus my first row.

00:03:08.150 --> 00:03:12.970
So minus 2 times
2 plus 1 is 0.

00:03:12.970 --> 00:03:16.270
1 times 2 plus minus 2 is 0.

00:03:16.270 --> 00:03:19.190
1 times 2 plus minus 2 is 0.

00:03:19.190 --> 00:03:20.900
This row is the same
as this row.

00:03:20.900 --> 00:03:22.180
So I'm going to do
the same thing.

00:03:22.180 --> 00:03:25.360
Minus 2 times 2 plus 4 is 0.

00:03:25.360 --> 00:03:27.860
1 times 2 plus 2 is 0.

00:03:27.860 --> 00:03:31.570
And then 1 times 2 plus
minus 2 is 0.

00:03:31.570 --> 00:03:33.840
So the solutions to this
equation are the same as the

00:03:33.840 --> 00:03:35.270
solutions to this equation.

00:03:35.270 --> 00:03:37.250
Let me write it like this.

00:03:37.250 --> 00:03:38.470
Instead of just writing
the vector, v,

00:03:38.470 --> 00:03:40.700
let me write it out.

00:03:40.700 --> 00:03:47.570
So v1, v2, v3 are going to
be equal to the 0 vector.

00:03:47.570 --> 00:03:48.300
0, 0.

00:03:48.300 --> 00:03:50.180
Just rewriting it slightly
different.

00:03:50.180 --> 00:03:53.010
And so these two rows, or these
two equations, give us

00:03:53.010 --> 00:03:53.700
no information.

00:03:53.700 --> 00:03:58.490
The only one is this row up
here, which tells us that 4

00:03:58.490 --> 00:04:04.560
times v1 minus 2 times v2--
actually this wasn't complete

00:04:04.560 --> 00:04:06.820
reduced row echelon form
but close enough.

00:04:06.820 --> 00:04:10.050
It's easy for us to work with--
4 times v1 minus 2

00:04:10.050 --> 00:04:17.920
times v2 minus 2 times
v3 is equal to 0.

00:04:17.920 --> 00:04:20.050
Let's just divide by 4.

00:04:20.050 --> 00:04:22.830
I could've just divided by 4
here, which might have made it

00:04:22.830 --> 00:04:23.970
skipped a step.

00:04:23.970 --> 00:04:30.210
But if you divide by 4 you get
v1 minus 1/2 v2 minus 1/2 v3

00:04:30.210 --> 00:04:31.620
is equal to 0.

00:04:31.620 --> 00:04:36.490
Or, v1 is equal to 1/2
v2 plus 1/2 v3.

00:04:36.490 --> 00:04:39.400
Just added these guys to both
sides of the equation.

00:04:39.400 --> 00:04:45.770
Or we could say, let's say that
v2 is equal to-- yeah I

00:04:45.770 --> 00:04:50.180
don't know, I'm going to just
put some random number-- a,

00:04:50.180 --> 00:04:55.830
and v3 is equal to b, then we
can say-- and then v1 would be

00:04:55.830 --> 00:05:00.200
equal to 1/2 a plus 1/2 b.

00:05:00.200 --> 00:05:07.020
We can say that the eigenspace
for lambda is equal to 3, is

00:05:07.020 --> 00:05:14.900
the set of all of vectors, v1,
v2, v3, that are equal to a

00:05:14.900 --> 00:05:18.350
times times-- v2 is a, right?

00:05:18.350 --> 00:05:21.200
So v2 is equal to a times 1.

00:05:21.200 --> 00:05:22.810
v3 has no a in it.

00:05:22.810 --> 00:05:26.020
So it's a times 0.

00:05:26.020 --> 00:05:30.950
Plus b times-- v2 is just a.

00:05:30.950 --> 00:05:32.490
v2 has no b in it.

00:05:32.490 --> 00:05:33.640
So it's 0.

00:05:33.640 --> 00:05:39.290
v3 is 1 times-- so 0 times
a plus 1 times b.

00:05:39.290 --> 00:05:43.745
And then v1 is 1/2
a plus 1/2 b.

00:05:48.260 --> 00:05:52.800
For any a and b, such
that a and b are

00:05:52.800 --> 00:05:54.600
members of the reals.

00:05:54.600 --> 00:05:56.660
Just to be a little bit
formal about it.

00:05:56.660 --> 00:06:02.180
So that's our-- any vector
that satisfies this is an

00:06:02.180 --> 00:06:03.320
eigenvector.

00:06:03.320 --> 00:06:05.260
And they're the eigenvectors
that correspond to eigenvalue

00:06:05.260 --> 00:06:07.190
lambda is equal to 3.

00:06:07.190 --> 00:06:10.200
So if you apply the matrix
transformation to any of these

00:06:10.200 --> 00:06:14.450
vectors, you're just going
to scale them up by 3.

00:06:14.450 --> 00:06:16.800
Let me write this way.

00:06:16.800 --> 00:06:20.220
The eigenspace for lambda is
equal to 3, is equal to the

00:06:20.220 --> 00:06:23.950
span, all of the potential
linear combinations of this

00:06:23.950 --> 00:06:25.290
guy and that guy.

00:06:25.290 --> 00:06:28.710
So 1/2, 1, 0.

00:06:28.710 --> 00:06:36.430
And 1/2, 0, 1.

00:06:36.430 --> 00:06:39.640
So that's only one of
the eigenspaces.

00:06:39.640 --> 00:06:40.750
That's the one that
corresponds to

00:06:40.750 --> 00:06:41.640
lambda is equal to 3.

00:06:41.640 --> 00:06:43.290
Let's do the one that
corresponds to lambda is equal

00:06:43.290 --> 00:06:45.060
to minus 3.

00:06:45.060 --> 00:06:47.420
So if lambda is equal to minus
3-- I'll do it up here, I

00:06:47.420 --> 00:06:50.410
think I have enough space--
lambda is equal to minus 3.

00:06:50.410 --> 00:06:57.850
This matrix becomes-- I'll do
the diagonals-- minus 3 plus 1

00:06:57.850 --> 00:06:59.300
is minus 2.

00:06:59.300 --> 00:07:02.950
Minus 3 minus 2 is minus 5.

00:07:02.950 --> 00:07:05.960
Minus 3 minus 2 is minus 5.

00:07:05.960 --> 00:07:08.160
And all the other things
don't change.

00:07:08.160 --> 00:07:11.520
Minus 2, minus 2, 1.

00:07:11.520 --> 00:07:15.110
Minus 2, minus 2 and 1.

00:07:15.110 --> 00:07:20.210
And then that times vectors
in the eigenspace that

00:07:20.210 --> 00:07:24.390
corresponds to lambda is equal
to minus 3, is going to be

00:07:24.390 --> 00:07:25.050
equal to 0.

00:07:25.050 --> 00:07:27.140
I'm just applying this equation
right here which we

00:07:27.140 --> 00:07:29.550
just derived from that
one over there.

00:07:29.550 --> 00:07:34.116
So, the eigenspace that
corresponds to lambda is equal

00:07:34.116 --> 00:07:37.315
to minus 3, is the null space,
this matrix right here, are

00:07:37.315 --> 00:07:40.040
all the vectors that satisfy
this equation.

00:07:40.040 --> 00:07:42.210
So what is-- the null space of
this is the same thing as the

00:07:42.210 --> 00:07:45.670
null space of this in reduced
row echelon form So let's put

00:07:45.670 --> 00:07:48.190
it in reduced row
echelon form.

00:07:48.190 --> 00:07:51.510
So the first thing I want to do,
I'm going to keep my first

00:07:51.510 --> 00:07:52.360
row the same.

00:07:52.360 --> 00:07:54.620
I'm going to write a little bit
smaller than I normally do

00:07:54.620 --> 00:07:56.800
because I think I'm going
to run out of space.

00:07:56.800 --> 00:08:01.010
So minus 2, minus 2, minus 2.

00:08:01.010 --> 00:08:03.170
Actually let me just
do it this way.

00:08:03.170 --> 00:08:04.830
I will skip some steps.

00:08:04.830 --> 00:08:07.000
Let's just divide the first
row by minus 2.

00:08:07.000 --> 00:08:10.230
So we get 1, 1, 1.

00:08:10.230 --> 00:08:14.190
And then let's replace this
second row with the second row

00:08:14.190 --> 00:08:16.470
plus this version of
the first row.

00:08:16.470 --> 00:08:22.170
So this guy plus that guy is 0
minus 5 plus minus-- or let me

00:08:22.170 --> 00:08:22.960
say this way.

00:08:22.960 --> 00:08:27.350
Let me replace it with
the first row

00:08:27.350 --> 00:08:28.720
minus the second row.

00:08:28.720 --> 00:08:32.179
So minus 2 minus minus 2 is 0.

00:08:32.179 --> 00:08:36.450
Minus 2 minus minus
5 is plus 3.

00:08:36.450 --> 00:08:43.669
And then minus 2 minus
1 is minus 3.

00:08:43.669 --> 00:08:44.770
And then let me do
the last row in a

00:08:44.770 --> 00:08:46.060
different color for fun.

00:08:46.060 --> 00:08:47.330
And I'll do the same thing.

00:08:47.330 --> 00:08:49.840
I'll do this row
minus this row.

00:08:49.840 --> 00:08:54.110
So minus 2 minus
minus 2 is a 0.

00:08:54.110 --> 00:08:55.150
Minus 2 plus 2.

00:08:55.150 --> 00:08:58.390
Minus 2 minus 1 is minus 3.

00:08:58.390 --> 00:09:03.210
And then we have minus
2 minus minus 5.

00:09:03.210 --> 00:09:04.330
So it's minus 2 plus 5.

00:09:04.330 --> 00:09:06.090
So that is 3.

00:09:06.090 --> 00:09:13.770
Now let me replace-- and I'll
do it in two steps.

00:09:13.770 --> 00:09:15.570
So this is 1, 1, 1.

00:09:15.570 --> 00:09:18.770
I'll just keep it like that.

00:09:18.770 --> 00:09:23.480
And actually, well let me
just keep it like that.

00:09:23.480 --> 00:09:26.590
And then let me replace my third
row with my third row

00:09:26.590 --> 00:09:27.870
plus my second row.

00:09:27.870 --> 00:09:28.660
It'll just zero out.

00:09:28.660 --> 00:09:31.270
If you add these terms, these
all just become 0.

00:09:31.270 --> 00:09:32.540
That guy got zeroed out.

00:09:32.540 --> 00:09:35.410
And let me take my second
row and divide it by 3.

00:09:35.410 --> 00:09:39.530
So this becomes 0, 1, minus 1.

00:09:39.530 --> 00:09:42.970
And I'm almost there.

00:09:42.970 --> 00:09:45.330
I'll do it in orange.

00:09:45.330 --> 00:09:48.680
So let me replace my first row
with my first row minus my

00:09:48.680 --> 00:09:49.470
second row.

00:09:49.470 --> 00:09:57.250
So this becomes 1, 0, and then
1 minus minus 1 is 2.

00:09:57.250 --> 00:09:59.440
1 minus minus 1 is 2.

00:09:59.440 --> 00:10:03.760
And then in the second
row is 0, 1, minus 1.

00:10:03.760 --> 00:10:07.770
And then the last
row is 0, 0, 0.

00:10:07.770 --> 00:10:10.910
So any v that satisfies this
equation will also

00:10:10.910 --> 00:10:13.480
satisfy this guy.

00:10:13.480 --> 00:10:15.650
This guy's null space is going
to be the null space of that

00:10:15.650 --> 00:10:18.180
guy in reduced row
echelon form.

00:10:18.180 --> 00:10:26.010
So v1, v2, v3 is equal
to 0, 0, 0.

00:10:26.010 --> 00:10:26.930
Let me move this.

00:10:26.930 --> 00:10:29.530
Because I've officially
run out of space.

00:10:29.530 --> 00:10:33.110
So let me move this lower
down where I have

00:10:33.110 --> 00:10:35.620
some free real estate.

00:10:35.620 --> 00:10:36.800
Let me move it down here.

00:10:36.800 --> 00:10:41.430
This corresponds to lambda
is equal to minus 3.

00:10:41.430 --> 00:10:44.830
This was lambda is equal to
minus 3, just to make us--

00:10:44.830 --> 00:10:47.200
it's not related to this
stuff right here.

00:10:47.200 --> 00:10:51.620
So what are all of the v1s, v2s
and v3s that satisfy this?

00:10:51.620 --> 00:11:00.100
So if we say that v3
is equal to t.

00:11:00.100 --> 00:11:04.310
If v3 is equal to t, then
what do we have here?

00:11:04.310 --> 00:11:08.530
We have-- this tells us that
v2 minus v3 is equal to 0.

00:11:08.530 --> 00:11:15.700
So that tells us that v2 minus
v3-- 0 times v1 plus v2 minus

00:11:15.700 --> 00:11:18.150
v3 is equal to 0.

00:11:18.150 --> 00:11:22.660
Or that v2 is equal to v3,
which is equal to t.

00:11:22.660 --> 00:11:25.340
That's what that second
equation tells us.

00:11:25.340 --> 00:11:28.150
And then the third equation
tells us, or the top equation

00:11:28.150 --> 00:11:34.060
tells us, v1 times 1-- so v1
plus 0 times v2 plus 2 times

00:11:34.060 --> 00:11:37.990
v3 is equal to 0.

00:11:37.990 --> 00:11:45.240
Or v1 is equal to minus 2v3 is
equal to minus 2 times t.

00:11:45.240 --> 00:11:50.110
So the eigenspace that
corresponds to lambda is equal

00:11:50.110 --> 00:11:56.930
to minus 3 is equal to the set
of all the vectors, v1, v2 and

00:11:56.930 --> 00:12:07.820
v3, where-- well, it's equal
to t times-- v3 is just t.

00:12:07.820 --> 00:12:09.790
v3 was just t.

00:12:09.790 --> 00:12:12.400
v2 also just ends up being t.

00:12:12.400 --> 00:12:13.320
So 1 times t.

00:12:13.320 --> 00:12:17.530
And v1 is minus 2 times t.

00:12:17.530 --> 00:12:20.390
For t is any real number.

00:12:20.390 --> 00:12:25.030
Or another way to say it is that
the eigenspace for lambda

00:12:25.030 --> 00:12:31.380
is equal to minus 3 is equal
to the span-- I wrote this

00:12:31.380 --> 00:12:36.200
really messy-- where lambda is
equal to minus 3 is equal to

00:12:36.200 --> 00:12:45.020
the span of the vector
minus 2, 1, and 1.

00:12:45.020 --> 00:12:46.540
Just like that.

00:12:46.540 --> 00:12:47.620
It looks interesting.

00:12:47.620 --> 00:12:50.510
Because if you take this guy
and dot it with either of

00:12:50.510 --> 00:12:52.280
these guys, I think you get 0.

00:12:52.280 --> 00:12:54.510
Is that definitely the case?

00:12:54.510 --> 00:12:59.790
Take minus 2 times 1/2, you
get a minus 1 there.

00:12:59.790 --> 00:13:00.750
Then you have a plus 1.

00:13:00.750 --> 00:13:01.580
That's 0.

00:13:01.580 --> 00:13:03.750
And then minus 2 times 1/2.

00:13:03.750 --> 00:13:04.160
Yeah.

00:13:04.160 --> 00:13:06.230
You dot it with either of
these guys you get 0.

00:13:06.230 --> 00:13:09.200
So this line is orthogonal
to that plane.

00:13:09.200 --> 00:13:09.790
Very interesting.

00:13:09.790 --> 00:13:12.990
So let's just graph it just so
we have a good visualization

00:13:12.990 --> 00:13:13.770
of what we're doing.

00:13:13.770 --> 00:13:16.070
So we had that 3
by 3 matrix, A.

00:13:16.070 --> 00:13:18.920
It represents some
transformation in R3.

00:13:18.920 --> 00:13:21.210
And it has two eigenvalues.

00:13:21.210 --> 00:13:24.030
And each of those have a
corresponding eigenspace.

00:13:24.030 --> 00:13:26.030
So the eigenspace that
corresponds to the eigenvalue

00:13:26.030 --> 00:13:27.930
3 is a plane in R3.

00:13:31.940 --> 00:13:37.600
So this is the eigenspace for
lambda is equal to 3.

00:13:37.600 --> 00:13:40.370
And it's the span of these
two vectors right there.

00:13:40.370 --> 00:13:43.040
So if I draw them, maybe
they're like that.

00:13:43.040 --> 00:13:44.040
Just like that.

00:13:44.040 --> 00:13:46.440
And then the eigenspace
for lambda is equal to

00:13:46.440 --> 00:13:47.850
minus 3 is a line.

00:13:47.850 --> 00:13:50.260
It's a line that's perpendicular
to this plane.

00:13:50.260 --> 00:13:52.460
It's a line like that.

00:13:52.460 --> 00:13:53.710
It's the span of this guy.

00:13:53.710 --> 00:13:55.870
Maybe if I draw that vector,
that vector might look

00:13:55.870 --> 00:13:57.150
something like this.

00:13:57.150 --> 00:13:59.440
And it's the span of that guy.

00:13:59.440 --> 00:14:04.920
So what this tells us, this is
the eigenspace for lambda is

00:14:04.920 --> 00:14:06.600
equal to minus 3.

00:14:06.600 --> 00:14:08.830
So what that tells us-- just
to make sure we are

00:14:08.830 --> 00:14:12.300
interpreting our eigenvalues and
eigenspaces correctly-- is

00:14:12.300 --> 00:14:15.700
look, you give me any
eigenvector, you give me any

00:14:15.700 --> 00:14:18.960
vector in this, you give me any
vector right here, let's

00:14:18.960 --> 00:14:20.800
say that is vector x.

00:14:20.800 --> 00:14:24.320
If I apply the transformation,
if I multiply it it by a, I'm

00:14:24.320 --> 00:14:25.980
going to have 3 times that.

00:14:25.980 --> 00:14:29.440
Because it's in the eigenspace
where lambda is equal to 3.

00:14:29.440 --> 00:14:33.420
So if I were to apply a times
x, a times x would be just 3

00:14:33.420 --> 00:14:34.460
times that.

00:14:34.460 --> 00:14:36.220
So that would be a times x.

00:14:36.220 --> 00:14:36.900
That's what it tells me.

00:14:36.900 --> 00:14:38.890
This would be true for
any of these guys.

00:14:38.890 --> 00:14:41.390
If this was x, and you took a
times x, it's going to be 3

00:14:41.390 --> 00:14:42.740
times as long.

00:14:42.740 --> 00:14:47.950
Now these guys over here, if you
have some vector in this

00:14:47.950 --> 00:14:50.520
eigenspace that corresponds to
lambda is equal to 3, and you

00:14:50.520 --> 00:14:51.640
apply the transformation.

00:14:51.640 --> 00:14:53.500
Let's say that this
is x right there.

00:14:53.500 --> 00:14:55.710
If you took the transformation
of x, it's going to make it 3

00:14:55.710 --> 00:14:57.320
times longer in the opposite
direction.

00:14:57.320 --> 00:14:59.230
It's still going to
be on this line.

00:14:59.230 --> 00:15:01.720
So it's going to go
down like this.

00:15:01.720 --> 00:15:03.340
And that would be a times x.

00:15:03.340 --> 00:15:05.510
It would be the same, it'd be
3 times this length, but in

00:15:05.510 --> 00:15:06.360
the opposite direction.

00:15:06.360 --> 00:15:11.190
Because it corresponds to lambda
is equal to minus 3.

00:15:11.190 --> 00:15:14.420
So anyway, we've, I think,
made a great achievement.

00:15:14.420 --> 00:15:18.290
We've not only figured out the
eigenvalues for a 3 by 3

00:15:18.290 --> 00:15:20.760
matrix, we now have figured out
all of the eigenvectors.

00:15:20.760 --> 00:15:22.260
Which are-- there's an infinite
number-- but they

00:15:22.260 --> 00:15:26.590
represent 2 eigenspaces that
correspond to those two

00:15:26.590 --> 00:15:31.170
eigenvalues, or minus 3 and 3.

00:15:31.170 --> 00:15:33.270
See you in the next video.


Eigenvalues of a 3x3 matrix

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:04.230
We figured out the eigenvalues
for a 2 by 2 matrix, so let's

00:00:04.230 --> 00:00:06.240
see if we can figure
out the eigenvalues

00:00:06.240 --> 00:00:07.920
for a 3 by 3 matrix.

00:00:07.920 --> 00:00:10.730
And I think we'll appreciate
that it's a good bit more

00:00:10.730 --> 00:00:14.420
difficult just because the math
becomes a little hairier.

00:00:14.420 --> 00:00:22.830
So lambda is an eigenvalue
of A.

00:00:22.830 --> 00:00:26.810
By definition, if and only if--
I'll write it like this.

00:00:26.810 --> 00:00:33.180
If and only if A times some
non-zero vector v is equal to

00:00:33.180 --> 00:00:36.075
lambda times that non-zero
vector v.

00:00:36.075 --> 00:00:41.040
Let we write that for
some non-zero.

00:00:43.790 --> 00:00:46.930
I could call it eigenvector v,
but I'll just call it for some

00:00:46.930 --> 00:00:50.820
non-zero vector v or
some non-zero v.

00:00:50.820 --> 00:00:56.480
Now this is true if and only if,
this leads to-- I'll write

00:00:56.480 --> 00:00:58.220
it like this.

00:00:58.220 --> 00:01:00.520
This is true if and only if--
and this is a bit of review,

00:01:00.520 --> 00:01:02.880
but I like to review it just
because when you do this 10

00:01:02.880 --> 00:01:04.760
years from now, I don't want you
to remember the formula.

00:01:04.760 --> 00:01:07.230
I want you to just remember the
logic of how we got to it.

00:01:07.230 --> 00:01:10.910
So this is true if and only if--
let's just subtract Av

00:01:10.910 --> 00:01:15.270
from both sides-- the 0 vector
is equal to lambda- instead of

00:01:15.270 --> 00:01:17.840
writing lambda times v, I'm
going to write lambda times

00:01:17.840 --> 00:01:20.340
the identity matrix times v.

00:01:20.340 --> 00:01:21.580
This is the same thing.

00:01:21.580 --> 00:01:23.890
The identity matrix
times v is just v.

00:01:23.890 --> 00:01:26.660
Minus Av.

00:01:26.660 --> 00:01:29.280
I just subtracted Av from both
sides, rewrote v as the

00:01:29.280 --> 00:01:31.140
identity matrix times v.

00:01:31.140 --> 00:01:37.760
Well this is only true if and
only if the 0 vector is equal

00:01:37.760 --> 00:01:44.630
to lambda times the identity
matrix minus A times v.

00:01:44.630 --> 00:01:47.740
I just factored the vector v out
from the right-hand side

00:01:47.740 --> 00:01:49.540
of both of these guys, and
I'm just left with some

00:01:49.540 --> 00:01:51.700
matrix times v.

00:01:51.700 --> 00:01:55.900
Well this is only true-- let
me rewrite this over here,

00:01:55.900 --> 00:01:58.110
this equation just in a form
you might recognize it.

00:01:58.110 --> 00:02:01.110
Lambda times the identity
matrix times A.

00:02:01.110 --> 00:02:02.330
This is just some matrix.

00:02:02.330 --> 00:02:09.000
This matrix times v has got
to be equal to 0 for some

00:02:09.000 --> 00:02:10.910
non-zero vector v.

00:02:10.910 --> 00:02:16.020
That means that the null space
of this matrix has got to be

00:02:16.020 --> 00:02:18.070
nontrivial.

00:02:18.070 --> 00:02:21.880
Or another way to think about it
is that its columns are not

00:02:21.880 --> 00:02:23.310
linearly independent.

00:02:23.310 --> 00:02:25.800
Or another way to think about it
is it's not invertible, or

00:02:25.800 --> 00:02:27.550
it has a determinant of 0.

00:02:27.550 --> 00:02:31.250
So lambda is the eigenvalue of
A, if and only if, each of

00:02:31.250 --> 00:02:32.550
these steps are true.

00:02:32.550 --> 00:02:35.780
And this is true if and only
if-- for some at non-zero

00:02:35.780 --> 00:02:43.560
vector, if and only if, the
determinant of lambda times

00:02:43.560 --> 00:02:47.630
the identity matrix minus
A is equal to 0.

00:02:47.630 --> 00:02:48.560
And that was our takeaway.

00:02:48.560 --> 00:02:51.160
I think it was two videos
ago or three videos ago.

00:02:51.160 --> 00:02:56.620
But let's apply it now to
this 3 by 3 matrix A.

00:02:56.620 --> 00:02:59.680
We're going to use the 3
by 3 identity matrix.

00:02:59.680 --> 00:03:01.985
So we want to concern ourselves
with-- lambda times

00:03:01.985 --> 00:03:05.650
the identity matrix is just
going to be-- times the 3 by 3

00:03:05.650 --> 00:03:09.410
identity matrix is just
going to be-- this is,

00:03:09.410 --> 00:03:09.900
let me write this.

00:03:09.900 --> 00:03:13.920
This is lambda times the
identity matrix in R3.

00:03:13.920 --> 00:03:17.950
So it's just going to be
lambda, lambda, lambda.

00:03:17.950 --> 00:03:21.100
And everything else is
going to be 0's.

00:03:21.100 --> 00:03:23.400
The identity matrix had 1's
across here, so that's the

00:03:23.400 --> 00:03:25.200
only thing that becomes
non-zero when you

00:03:25.200 --> 00:03:26.230
multiply it by lambda.

00:03:26.230 --> 00:03:27.760
Everything else was a 0.

00:03:27.760 --> 00:03:32.690
So that's the identity
matrix times lambda.

00:03:32.690 --> 00:03:36.700
So lambda times the identity
matrix minus A is going to be

00:03:36.700 --> 00:03:41.230
equal to-- it's actually pretty

00:03:41.230 --> 00:03:42.550
straightforward to find.

00:03:42.550 --> 00:03:45.170
Everything along the diagonal is
going to be lambda minus--

00:03:45.170 --> 00:03:46.060
let's just do it.

00:03:46.060 --> 00:03:48.900
Lambda minus minus 1-- I'll
do the diagonals here.

00:03:48.900 --> 00:03:53.230
Lambda minus minus 1
is lambda plus 1.

00:03:53.230 --> 00:03:56.040
And then 0 minus 2-- I'll do
that in a different color.

00:03:56.040 --> 00:03:58.250
0 minus 2 is minus 2.

00:03:58.250 --> 00:04:01.190
0 minus 2 is minus 2.

00:04:01.190 --> 00:04:03.620
0 minus 2 is minus 2.

00:04:03.620 --> 00:04:04.270
Let's do this one.

00:04:04.270 --> 00:04:07.380
0 minus 2 is minus 2.

00:04:07.380 --> 00:04:12.460
0 plus or minus minus 1 is
0 plus 1, which is 1.

00:04:12.460 --> 00:04:13.560
And then let's just
do this one.

00:04:13.560 --> 00:04:15.460
0 minus minus 1.

00:04:15.460 --> 00:04:16.810
That's one.

00:04:16.810 --> 00:04:19.700
Let me finish up the diagonal.

00:04:19.700 --> 00:04:21.695
And then you have
lambda minus 2.

00:04:24.320 --> 00:04:27.900
And then you have
lambda minus 2.

00:04:27.900 --> 00:04:32.280
So lambda is an eigenvalue
of A if and only if the

00:04:32.280 --> 00:04:36.740
determinant of this matrix
right here is equal to 0.

00:04:36.740 --> 00:04:38.570
Let's figure out its
determinate.

00:04:38.570 --> 00:04:41.050
And the easiest way, at least
in my head to do this, is to

00:04:41.050 --> 00:04:43.025
use the rule of Sarrus.

00:04:43.025 --> 00:04:45.620
So let's use the rule of
Sarrus to find this

00:04:45.620 --> 00:04:46.650
determinant.

00:04:46.650 --> 00:04:49.460
So I just rewrite these
rows right there.

00:04:49.460 --> 00:04:52.070
I could just copy and
paste them really.

00:04:52.070 --> 00:04:55.640
I just take those two rows.

00:04:55.640 --> 00:04:59.070
And then let me paste them,
put them right there.

00:04:59.070 --> 00:05:01.720
It's a little bit too close
to this guy, but I

00:05:01.720 --> 00:05:03.310
think you get the idea.

00:05:03.310 --> 00:05:08.130
And now the rule of Sarrus I
just take this product plus

00:05:08.130 --> 00:05:13.090
this product plus this product
and then I subtract out this

00:05:13.090 --> 00:05:15.110
product times this product
times this product.

00:05:15.110 --> 00:05:16.570
We'll do that next.

00:05:16.570 --> 00:05:24.220
So this product is lambda plus
1 times lambda minus 2 times

00:05:24.220 --> 00:05:26.020
lambda minus 2.

00:05:26.020 --> 00:05:27.400
That's that one there.

00:05:27.400 --> 00:05:30.470
And then plus, let's see,
minus 2 times minus 2.

00:05:30.470 --> 00:05:32.216
That's plus 4.

00:05:32.216 --> 00:05:34.640
And then we have minus 2 times
minus 2 plus 4 times 1.

00:05:34.640 --> 00:05:37.790
So that is plus 4 again.

00:05:37.790 --> 00:05:43.910
And then we do minus this column
times this column.

00:05:43.910 --> 00:05:46.840
Minus this column minus this
column and then-- or I

00:05:46.840 --> 00:05:49.210
shouldn't say column,
but diagonal really.

00:05:49.210 --> 00:05:52.890
So we say minus 2
times minus 2.

00:05:52.890 --> 00:05:55.750
Let me write this.

00:05:55.750 --> 00:05:59.510
Minus 2 times minus
2, which is 4.

00:05:59.510 --> 00:06:03.670
Times lambda minus 2.

00:06:03.670 --> 00:06:05.490
That was this diagonal.

00:06:05.490 --> 00:06:08.970
And then we have minus-- what
is this going to be?

00:06:08.970 --> 00:06:11.510
Going to be minus 1 times
lambda plus 1.

00:06:11.510 --> 00:06:14.840
So minus lambda plus 1.

00:06:14.840 --> 00:06:17.010
And then you go down
this diagonal.

00:06:17.010 --> 00:06:19.410
Minus 2 times minus 2 is 4.

00:06:19.410 --> 00:06:21.410
So it's going to be 4 times
lambda minus 2 and we're

00:06:21.410 --> 00:06:21.840
subtracting.

00:06:21.840 --> 00:06:25.750
So minus 4 times
lambda minus 2.

00:06:25.750 --> 00:06:28.200
And let's see if we
can simplify this.

00:06:28.200 --> 00:06:31.660
So this blue stuff over here--
let's see, these guys right

00:06:31.660 --> 00:06:36.230
here become an 8 and then
this becomes-- this

00:06:36.230 --> 00:06:39.510
becomes lambda plus 1.

00:06:39.510 --> 00:06:43.550
Times-- if I multiply these two
guys out, lambda squared

00:06:43.550 --> 00:06:44.860
minus 4 lambda.

00:06:44.860 --> 00:06:46.440
Minus 2 lambda and then
minus 2 lambda.

00:06:46.440 --> 00:06:48.690
So minus 4 lambda.

00:06:48.690 --> 00:06:50.200
Plus 4.

00:06:50.200 --> 00:06:54.190
And then I have this
plus 8 here.

00:06:54.190 --> 00:06:55.130
And then I have-- let's see.

00:06:55.130 --> 00:06:56.900
I have minus 4 times lambda.

00:06:56.900 --> 00:06:58.260
Let me just multiply
everything out.

00:06:58.260 --> 00:07:07.900
So I have minus 4 lambda plus 8
minus lambda minus 1 minus 4

00:07:07.900 --> 00:07:14.450
lambda plus 8.

00:07:14.450 --> 00:07:17.800
And then let me simplify
this up a little bit.

00:07:17.800 --> 00:07:19.310
So this guy over here--
let's see.

00:07:19.310 --> 00:07:24.305
The constant terms, I have an 8,
I have a minus 1, I have an

00:07:24.305 --> 00:07:26.220
8 and I have an 8.

00:07:26.220 --> 00:07:28.680
So that's 24 minus 1.

00:07:28.680 --> 00:07:31.910
So that is a 23.

00:07:31.910 --> 00:07:36.220
And then the lambda terms
I have a minus 4 lambda.

00:07:36.220 --> 00:07:39.090
I have a minus lambda and
I have a minus 4 lambda.

00:07:39.090 --> 00:07:41.320
So it's minus 8, minus 1.

00:07:41.320 --> 00:07:44.750
So I have minus 9 lambda.

00:07:44.750 --> 00:07:45.670
Plus 23.

00:07:45.670 --> 00:07:48.530
And now I have to simplify
this out.

00:07:48.530 --> 00:07:51.020
So first I can take lambda and
multiply it times this whole

00:07:51.020 --> 00:07:51.910
guy right there.

00:07:51.910 --> 00:08:00.000
So it's going to be lambda cubed
minus 4 lambda squared

00:08:00.000 --> 00:08:01.770
plus 4 lambda.

00:08:01.770 --> 00:08:03.895
And then I can take this
one and multiply

00:08:03.895 --> 00:08:05.400
it times that guy.

00:08:05.400 --> 00:08:09.440
So plus lambda squared.

00:08:09.440 --> 00:08:13.980
Minus 4 lambda plus 4.

00:08:13.980 --> 00:08:15.990
And now of course, we have
these terms over here.

00:08:15.990 --> 00:08:18.740
So we're going to have
to simplify it again.

00:08:18.740 --> 00:08:21.100
So what are all of our
constant terms?

00:08:21.100 --> 00:08:23.880
We have a 23 and we
have a plus 4.

00:08:23.880 --> 00:08:25.790
So we have a 27.

00:08:25.790 --> 00:08:27.350
Plus 27.

00:08:27.350 --> 00:08:29.220
And then, what are all
of our lambda terms?

00:08:29.220 --> 00:08:33.559
We have a minus 9 lambda and
then we have a-- let's see.

00:08:33.559 --> 00:08:37.090
We have a minus 9 lambda, we
have a plus 4 lambda, and then

00:08:37.090 --> 00:08:39.120
we have a minus 4 lambda.

00:08:39.120 --> 00:08:40.429
So these two cancel out.

00:08:40.429 --> 00:08:45.140
So I just have a
minus 9 lambda.

00:08:45.140 --> 00:08:47.840
And then, what are my lambda
squared terms?

00:08:47.840 --> 00:08:51.300
I have a plus lambda squared
and I have a

00:08:51.300 --> 00:08:54.190
minus 4 lambda squared.

00:08:54.190 --> 00:08:55.500
So if you add those two
that's going to be

00:08:55.500 --> 00:08:57.025
minus 3 lambda squared.

00:09:00.720 --> 00:09:04.130
And then finally, I have only
one lambda cubed term, that

00:09:04.130 --> 00:09:05.670
right there.

00:09:05.670 --> 00:09:10.450
So this is the characteristic
polynomial for our matrix.

00:09:10.450 --> 00:09:15.170
So this is the characteristic
polynomial and this represents

00:09:15.170 --> 00:09:18.350
the determinant for
any lambda.

00:09:18.350 --> 00:09:21.410
The determinant of this
matrix for any lambda.

00:09:21.410 --> 00:09:25.210
And we said that this has to be
equal to 0 if any only if

00:09:25.210 --> 00:09:26.820
lambda is truly an eigenvalue.

00:09:26.820 --> 00:09:31.260
So we're going to set
this equal to 0.

00:09:31.260 --> 00:09:35.570
And unlucky or lucky for us,
there is no real trivial--

00:09:35.570 --> 00:09:36.380
there is no quadratic.

00:09:36.380 --> 00:09:39.990
Well there is, actually, but
it's very complicated.

00:09:39.990 --> 00:09:41.600
And so it's usually
a waste of time.

00:09:41.600 --> 00:09:44.300
So we're going to have to do
kind of the art of factoring a

00:09:44.300 --> 00:09:45.440
quadratic polynomial.

00:09:45.440 --> 00:09:48.270
I got this problem out of a book
and I think it's fair to

00:09:48.270 --> 00:09:51.020
say that if you ever do run into
this in an actual linear

00:09:51.020 --> 00:09:52.970
algebra class or really, in an
algebra class generally-- it

00:09:52.970 --> 00:09:55.380
doesn't even have to be in the
context of eigenvalues, you

00:09:55.380 --> 00:09:57.920
probably will be dealing
with integer solutions.

00:09:57.920 --> 00:10:03.740
And if you are dealing with
integer solutions, then your

00:10:03.740 --> 00:10:07.540
roots are going to be factors
of this term right here.

00:10:07.540 --> 00:10:09.960
Especially if you have a
1 coefficient out here.

00:10:09.960 --> 00:10:12.140
So your potential roots-- in
this case, what are the

00:10:12.140 --> 00:10:13.145
factors of 27?

00:10:13.145 --> 00:10:18.720
So 1, 3, 9 and 27.

00:10:18.720 --> 00:10:20.230
So all these are potential
roots.

00:10:20.230 --> 00:10:21.410
So we can just try them out.

00:10:21.410 --> 00:10:24.280
1 cubed is 1 minus 3.

00:10:24.280 --> 00:10:25.900
So let me try 1.

00:10:25.900 --> 00:10:31.300
So if we try a 1, it's 1 minus
3 minus 9 plus 27.

00:10:31.300 --> 00:10:33.630
That does not equal 0.

00:10:33.630 --> 00:10:36.640
It's minus 2 minus
9 is minus 11.

00:10:36.640 --> 00:10:37.510
Plus 16.

00:10:37.510 --> 00:10:38.580
That does not equal 0.

00:10:38.580 --> 00:10:40.490
So 1 is not a root.

00:10:40.490 --> 00:10:46.060
If we try 3 we get 3
cubed, which is 27.

00:10:46.060 --> 00:10:48.960
Minus 3 times 3 squared
is minus 3 times 3,

00:10:48.960 --> 00:10:50.800
which is minus 27.

00:10:50.800 --> 00:10:54.320
Minus 9 times 3, which
is minus 27.

00:10:54.320 --> 00:10:55.470
Plus 27.

00:10:55.470 --> 00:10:56.820
That does equal 0.

00:10:56.820 --> 00:10:59.450
So lucky for us, on our second
try we were able to

00:10:59.450 --> 00:11:01.790
find one 0 for this.

00:11:01.790 --> 00:11:06.070
So if 3 is a 0, that means that
x minus 3 is one of the

00:11:06.070 --> 00:11:07.550
factors of this.

00:11:07.550 --> 00:11:10.310
So that means that this is going
to be x minus 3 times

00:11:10.310 --> 00:11:11.070
something else.

00:11:11.070 --> 00:11:13.010
Or I should say,
lambda minus 3.

00:11:13.010 --> 00:11:15.240
So let's see what the
other root is.

00:11:15.240 --> 00:11:22.310
So if I take lambda minus 3 and
I divide it into this guy

00:11:22.310 --> 00:11:30.250
up here, into lambda cubed minus
3 lambda squared minus 9

00:11:30.250 --> 00:11:34.870
lambda plus 27, what do I get?

00:11:34.870 --> 00:11:38.800
Lambda goes into lambda cubed
lambda squared times.

00:11:38.800 --> 00:11:40.220
Lambda squared times that.

00:11:40.220 --> 00:11:42.650
Lambda squared times lambda
is lambda cubed.

00:11:42.650 --> 00:11:48.710
Lambda squared times minus 3
is minus 3 lambda squared.

00:11:48.710 --> 00:11:53.280
You subtract these guys,
you get a 0.

00:11:53.280 --> 00:11:54.270
You get 0.

00:11:54.270 --> 00:11:57.100
And then we can put here--
well, we could

00:11:57.100 --> 00:11:57.810
do it either way.

00:11:57.810 --> 00:12:01.030
We could put it down
the minus 9.

00:12:01.030 --> 00:12:03.690
We could bring down
everything really.

00:12:03.690 --> 00:12:09.980
So now you have minus
9 lambda plus 27.

00:12:09.980 --> 00:12:11.910
You can almost imagine we just
subtracted this from this

00:12:11.910 --> 00:12:12.960
whole thing up here.

00:12:12.960 --> 00:12:15.480
And we're just left with
these terms right here.

00:12:15.480 --> 00:12:18.380
And so lambda minus
3 goes into this.

00:12:18.380 --> 00:12:23.160
Well lambda minus 3 goes
into 9 lambda.

00:12:23.160 --> 00:12:27.110
It goes into 9 lambda
minus 9 times.

00:12:27.110 --> 00:12:30.160
So I'll just write
minus 9 here.

00:12:30.160 --> 00:12:38.820
Minus 9 times lambda minus 3
is minus 9 lambda plus 27.

00:12:38.820 --> 00:12:41.550
So it went in very nicely.

00:12:41.550 --> 00:12:45.650
So you get to 0.

00:12:45.650 --> 00:12:53.150
Our characteristic polynomial
has simplified to lambda minus

00:12:53.150 --> 00:12:58.730
3 times lambda squared
minus 9.

00:12:58.730 --> 00:13:01.070
And of course, we're going to
have to set this equal to 0 if

00:13:01.070 --> 00:13:03.850
lambda is truly an eigenvalue
of our matrix.

00:13:03.850 --> 00:13:05.890
And this is very
easy to factor.

00:13:05.890 --> 00:13:12.770
So this becomes lambda minus 3
times-- lambda squared minus 9

00:13:12.770 --> 00:13:18.310
is just lambda plus 3 times
lambda minus 3.

00:13:18.310 --> 00:13:19.870
And all of that equals 0.

00:13:19.870 --> 00:13:21.650
And these roots, we already
know one of them.

00:13:21.650 --> 00:13:24.010
We know that 3 is a root and
actually, this tells us 3 is a

00:13:24.010 --> 00:13:24.980
root as well.

00:13:24.980 --> 00:13:30.500
So the possible eigenvalues of
our matrix A, our 3 by 3

00:13:30.500 --> 00:13:33.960
matrix A that we had way up
there-- this matrix A right

00:13:33.960 --> 00:13:38.040
there-- the possible eigenvalues
are: lambda is

00:13:38.040 --> 00:13:44.410
equal to 3 or lambda is
equal to minus 3.

00:13:44.410 --> 00:13:47.140
Those are the two values that
would make our characteristic

00:13:47.140 --> 00:13:52.710
polynomial or the determinant
for this matrix equal to 0,

00:13:52.710 --> 00:13:56.120
which is a condition that we
need to have in order for

00:13:56.120 --> 00:14:02.540
lambda to be an eigenvalue of a
for some non-zero vector v.

00:14:02.540 --> 00:14:04.050
In the next video, we'll
actually solve for the

00:14:04.050 --> 00:14:06.840
eigenvectors, now that we know
what the eigenvalues are.


Finding eigenvectors and eigenspaces example

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:05.140
In the last video, we started
with the 2 by 2 matrix A is

00:00:05.140 --> 00:00:09.440
equal to 1, 2, 4, 3.

00:00:09.440 --> 00:00:21.925
And we used the fact that lambda
is an eigenvalue of A,

00:00:21.925 --> 00:00:27.170
if and only if, the determinate
of lambda times

00:00:27.170 --> 00:00:30.560
the identity matrix-- in this
case it's a 2 by 2 identity

00:00:30.560 --> 00:00:33.360
matrix-- minus A
is equal to 0.

00:00:33.360 --> 00:00:36.380
This gave us a characteristic
polynomial and we solved for

00:00:36.380 --> 00:00:40.050
that and we said, well, the
eigenvalues for A are lambda

00:00:40.050 --> 00:00:45.090
is equal to 5 and lambda
is equal to negative 1.

00:00:45.090 --> 00:00:47.420
That's what we saw in
the last video.

00:00:47.420 --> 00:00:52.160
We said that if you were trying
to solve A times some

00:00:52.160 --> 00:00:56.160
eigenvector is equal to lambda
times that eigenvector, the

00:00:56.160 --> 00:00:59.790
two lambdas, which this equation
can be solved for,

00:00:59.790 --> 00:01:01.920
are the lambdas 5 and minus 1.

00:01:04.730 --> 00:01:09.725
Assuming nonzero eigenvectors.

00:01:18.110 --> 00:01:21.300
So we have our eigenvalues,
but I don't even call that

00:01:21.300 --> 00:01:22.110
half the battle.

00:01:22.110 --> 00:01:24.510
What we really want is our
eigenvectors and our

00:01:24.510 --> 00:01:24.850
eigenvalues.

00:01:24.850 --> 00:01:27.000
So let's see if we
can do that.

00:01:27.000 --> 00:01:30.580
So if we manipulate this
equation a little bit and

00:01:30.580 --> 00:01:32.320
we've manipulate it in the past.
Actually, we've even

00:01:32.320 --> 00:01:34.320
come up with this statement
over here.

00:01:34.320 --> 00:01:38.650
We can rewrite this over here
as the 0 vector is equal to

00:01:38.650 --> 00:01:43.180
lambda times my eigenvector
minus A times my eigenvector.

00:01:43.180 --> 00:01:45.780
I just subtracted Av
from both sides.

00:01:45.780 --> 00:01:48.860
We know lambda times some
eigenvector is the same thing

00:01:48.860 --> 00:01:54.330
as lambda times the identity
matrix times that eigenvector.

00:01:54.330 --> 00:01:58.040
So all I'm doing is rewriting
this like that.

00:01:58.040 --> 00:02:00.470
You multiply the identity matrix
times an eigenvector or

00:02:00.470 --> 00:02:02.230
times any vector, you're just
going to get that vector.

00:02:02.230 --> 00:02:03.950
So these two things
are equivalent.

00:02:03.950 --> 00:02:05.200
Minus Av.

00:02:07.750 --> 00:02:09.440
That's still going to be
able to the 0 vector.

00:02:09.440 --> 00:02:11.470
So far all I've done is
manipulated this thing.

00:02:11.470 --> 00:02:15.080
This is really how we got
to that thing up there.

00:02:15.080 --> 00:02:18.470
You factor out the v so to speak
because we know that

00:02:18.470 --> 00:02:21.710
matrix vector products exhibit
the distributive property.

00:02:21.710 --> 00:02:28.450
And we get lambda times the
identity matrix minus A times

00:02:28.450 --> 00:02:33.020
my eigenvector have got
to be equal to 0.

00:02:33.020 --> 00:02:43.960
Or another way to say it is, for
any lambda eigenvalue, and

00:02:43.960 --> 00:02:51.030
let me write it for any
eigenvalue lambda, the

00:02:51.030 --> 00:02:54.610
eigenvectors that correspond
to that lambda, we can call

00:02:54.610 --> 00:02:59.220
that the eigenspace
for a lambda.

00:02:59.220 --> 00:03:00.740
So that's a new word,
eigenspace.

00:03:04.560 --> 00:03:07.430
Eigenspace just means all
of the eigenvectors that

00:03:07.430 --> 00:03:09.950
correspond to some eigenvalue.

00:03:09.950 --> 00:03:16.040
The eigenspace for some
particular eigenvalue is going

00:03:16.040 --> 00:03:18.390
to be equal to the set
of vectors that

00:03:18.390 --> 00:03:19.810
satisfy this equation.

00:03:19.810 --> 00:03:22.790
Well, the set of vectors that
satisfy this equation is just

00:03:22.790 --> 00:03:26.450
the null space of that
right there.

00:03:26.450 --> 00:03:29.120
So it's equal to the
null space of this

00:03:29.120 --> 00:03:30.050
matrix right there.

00:03:30.050 --> 00:03:34.090
The null space of lambda times
the identity matrix.

00:03:34.090 --> 00:03:36.940
And by an identity
matrix minus A.

00:03:36.940 --> 00:03:39.040
And so everything I've done
here, this is true-- this is

00:03:39.040 --> 00:03:39.840
the general case.

00:03:39.840 --> 00:03:42.450
But now we can apply
this notion to this

00:03:42.450 --> 00:03:44.690
matrix A right here.

00:03:44.690 --> 00:03:48.840
So we know that 5 is
an eigenvalue.

00:03:48.840 --> 00:03:53.600
Let's say for lambda is equal
to 5, the eigenspace that

00:03:53.600 --> 00:03:59.140
corresponds to 5 is equal
to the null space of?

00:03:59.140 --> 00:04:03.510
Well, what is 5 times
the identity matrix?

00:04:03.510 --> 00:04:06.980
It's going to be the 2
by 2 identity matrix.

00:04:06.980 --> 00:04:15.730
5 times the identity matrix is
just 5, 0, 0, 5 minus A.

00:04:15.730 --> 00:04:19.959
That's just 1, 2, 4, 3.

00:04:19.959 --> 00:04:27.600
So that is equal to the null
space of the matrix.

00:04:27.600 --> 00:04:29.870
5 minus 1 is 4.

00:04:29.870 --> 00:04:32.640
0 minus 2 is minus 2.

00:04:32.640 --> 00:04:36.240
0 minus 4 is minus 4.

00:04:36.240 --> 00:04:40.090
And then, 5 minus 3 is 2.

00:04:40.090 --> 00:04:43.130
So the null space of this matrix
right here-- and this

00:04:43.130 --> 00:04:46.560
matrix is just an actual
numerical representation of

00:04:46.560 --> 00:04:48.150
this matrix right here.

00:04:48.150 --> 00:04:51.760
The null space of this matrix
is the set of all of the

00:04:51.760 --> 00:04:55.030
vectors that satisfy this or all
of the eigenvectors that

00:04:55.030 --> 00:04:56.950
correspond to this eigenvalue.

00:04:56.950 --> 00:04:59.100
Or, the eigenspace that
corresponds to

00:04:59.100 --> 00:05:00.590
the eigenvalue 5.

00:05:00.590 --> 00:05:02.540
These are all equivalent
statements.

00:05:02.540 --> 00:05:05.010
So we just need to figure out
the null space of this guy is

00:05:05.010 --> 00:05:09.200
all of the vectors that satisfy
the equation 4 minus

00:05:09.200 --> 00:05:13.790
2, minus 4, 2 times some
eigenvector is

00:05:13.790 --> 00:05:15.730
equal to the 0 vector.

00:05:15.730 --> 00:05:19.170
And the null space of a matrix
is equal to the null space of

00:05:19.170 --> 00:05:21.290
the reduced row echelon
form of a matrix.

00:05:21.290 --> 00:05:24.260
So what's the reduced row
echelon form of this guy?

00:05:24.260 --> 00:05:29.440
Well, I guess a good starting
point-- let me keep my first

00:05:29.440 --> 00:05:31.570
row the same, 4 minus 2.

00:05:31.570 --> 00:05:34.350
And let me replace my second
row with my second row

00:05:34.350 --> 00:05:35.590
plus my first row.

00:05:35.590 --> 00:05:38.040
So minus 4 plus 4 is 0.

00:05:38.040 --> 00:05:40.910
2 plus minus 2 is 0.

00:05:40.910 --> 00:05:44.850
Now, let me divide my
first row by 4 and I

00:05:44.850 --> 00:05:47.890
get 1, minus 1/2.

00:05:47.890 --> 00:05:49.810
And then I get 0, 0.

00:05:49.810 --> 00:05:51.340
So what's the null
space of this?

00:05:51.340 --> 00:05:53.280
This corresponds to v.

00:05:53.280 --> 00:05:58.550
This times v1, v2-- that's just
another way of writing my

00:05:58.550 --> 00:06:03.100
eigenvector v-- has got to
be equal to the 0 vector.

00:06:03.100 --> 00:06:08.560
Or another way to say it is that
my first entry v1, which

00:06:08.560 --> 00:06:12.810
corresponds to this pivot
column, plus or minus 1/2

00:06:12.810 --> 00:06:15.970
times my second entry has
got to be equal to

00:06:15.970 --> 00:06:17.730
that 0 right there.

00:06:17.730 --> 00:06:24.530
Or, v1 is equal to 1/2 v2.

00:06:24.530 --> 00:06:27.120
And so if I wanted to write all
of the eigenvectors that

00:06:27.120 --> 00:06:29.320
satisfy this, I could
write it this way.

00:06:29.320 --> 00:06:35.780
My eigenspace that corresponds
to lambda equals 5.

00:06:35.780 --> 00:06:39.360
That corresponds to the
eigenvalue 5 is equal to the

00:06:39.360 --> 00:06:48.250
set of all of the vectors, v1,
v2, that are equal to some

00:06:48.250 --> 00:06:49.310
scaling factor.

00:06:49.310 --> 00:06:53.360
Let's say it's equal
to t times what?

00:06:53.360 --> 00:06:58.170
If we say that v2 is equal to t,
so v2 is going to be equal

00:06:58.170 --> 00:07:01.140
to t times 1.

00:07:01.140 --> 00:07:05.550
And then, v1 is going to be
equal to 1/2 times v2

00:07:05.550 --> 00:07:06.850
or 1/2 times t.

00:07:09.390 --> 00:07:10.640
Just like that.

00:07:13.550 --> 00:07:16.150
For any t is a member
of the real numbers.

00:07:16.150 --> 00:07:17.850
If we wanted to, we could
scale this up.

00:07:17.850 --> 00:07:20.260
We could say any real
number times 1, 2.

00:07:20.260 --> 00:07:23.610
That would also be the span.

00:07:23.610 --> 00:07:24.500
Let me do that actually.

00:07:24.500 --> 00:07:26.060
It'll make it a little
bit cleaner.

00:07:26.060 --> 00:07:27.070
Actually, I don't
have to do that.

00:07:27.070 --> 00:07:32.820
So we could write that the
eigenspace for the eigenvalue

00:07:32.820 --> 00:07:42.060
5 is equal to the span of
the vector 1/2 and 1.

00:07:42.060 --> 00:07:44.030
So it's a line in R2.

00:07:44.030 --> 00:07:47.190
Those are all of the
eigenvectors that satisfy--

00:07:47.190 --> 00:07:48.700
that work for the equation
where the

00:07:48.700 --> 00:07:51.960
eigenvalue is equal to 5.

00:07:51.960 --> 00:07:53.620
Now what about when
the eigenvalue is

00:07:53.620 --> 00:07:55.080
equal to minus 1?

00:07:59.120 --> 00:08:00.750
So let's do that case.

00:08:00.750 --> 00:08:07.620
When lambda is equal to minus
1, then we have-- it's going

00:08:07.620 --> 00:08:09.660
to be the null space.

00:08:09.660 --> 00:08:13.370
So the eigenspace for lambda is
equal to minus 1 is going

00:08:13.370 --> 00:08:18.760
to be the null space of lambda
times our identity matrix,

00:08:18.760 --> 00:08:23.000
which is going to be minus
1 and 0, 0, minus 1.

00:08:23.000 --> 00:08:26.440
It's going to be minus 1 times
1, 0, 0, 1, which is just

00:08:26.440 --> 00:08:27.570
minus 1 there.

00:08:27.570 --> 00:08:28.800
Minus A.

00:08:28.800 --> 00:08:33.460
So minus 1, 2, 4, 3.

00:08:33.460 --> 00:08:37.620
And this is equal to the null
space of-- minus 1,

00:08:37.620 --> 00:08:40.760
minus 1 is minus 2.

00:08:40.760 --> 00:08:43.230
0 minus 2 is minus 2.

00:08:43.230 --> 00:08:49.160
0 minus 4 is minus 4 and minus
1 minus 3 is minus 4.

00:08:49.160 --> 00:08:51.270
And that's going to be equal
to the null space of the

00:08:51.270 --> 00:08:53.320
reduced row echelon
form of that guy.

00:08:53.320 --> 00:08:56.550
So we can perform some row
operations right here.

00:08:56.550 --> 00:08:59.730
Let me just put it in reduced
row echelon form.

00:08:59.730 --> 00:09:06.430
So if I replace my second row
plus 2 times my first row.

00:09:06.430 --> 00:09:08.070
So I'll keep the first
row the same.

00:09:08.070 --> 00:09:09.470
Minus 2, minus 2.

00:09:09.470 --> 00:09:11.550
And then my second row, I'll
replace it with two times--

00:09:11.550 --> 00:09:14.740
I'll replace it with it plus
2 times the first. Or even

00:09:14.740 --> 00:09:17.560
better, I'm going to replace it
with it plus minus 2 times

00:09:17.560 --> 00:09:20.025
the first. So minus
4 plus 4 is 0.

00:09:23.170 --> 00:09:26.940
And then if I divide the top
row by minus 2, the reduced

00:09:26.940 --> 00:09:30.110
row echelon form of this matrix
right here or this

00:09:30.110 --> 00:09:34.210
matrix right here is going
to be 1, 1, 0.

00:09:34.210 --> 00:09:38.140
So the eigenspace that
corresponds to the eigenvalue

00:09:38.140 --> 00:09:48.690
minus 1 is equal to the null
space of this guy right here

00:09:48.690 --> 00:09:50.930
It's the set of vectors
that satisfy this

00:09:50.930 --> 00:09:54.320
equation: 1, 1, 0, 0.

00:09:54.320 --> 00:09:59.110
And then you have v1,
v2 is equal to 0.

00:09:59.110 --> 00:10:02.720
Or you get v1 plus-- these
aren't vectors,

00:10:02.720 --> 00:10:04.070
these are just values.

00:10:04.070 --> 00:10:07.200
v1 plus v2 is equal to 0.

00:10:10.610 --> 00:10:14.020
Because 0 is just equal to
that thing right there.

00:10:14.020 --> 00:10:17.170
So 1 times v1 plus 1 times v2 is
going to be equal to that 0

00:10:17.170 --> 00:10:18.080
right there.

00:10:18.080 --> 00:10:21.910
Or I could write v1 is
equal to minus v2.

00:10:21.910 --> 00:10:28.510
Or if we say that v2 is equal to
t, we could say v1 is equal

00:10:28.510 --> 00:10:30.430
to minus t.

00:10:30.430 --> 00:10:34.450
Or we could say that the
eigenspace for the eigenvalue

00:10:34.450 --> 00:10:41.770
minus 1 is equal to all of the
vectors, v1, v2 that are equal

00:10:41.770 --> 00:10:50.180
to some scalar t times v1 is
minus t and v2 is plus t.

00:10:50.180 --> 00:10:54.410
Or you could say this is equal
to the span of the vector

00:10:54.410 --> 00:10:56.940
minus 1 and 1.

00:10:56.940 --> 00:10:59.710
So let's just graph this a
little bit just to understand

00:10:59.710 --> 00:11:00.960
what we just did.

00:11:00.960 --> 00:11:02.860
We were able to find
two eigenvalues for

00:11:02.860 --> 00:11:05.390
this, 5 and minus 1.

00:11:05.390 --> 00:11:08.370
And we were able to find all
of the vectors that are

00:11:08.370 --> 00:11:11.310
essentially-- or, we were able
to find the set of vectors

00:11:11.310 --> 00:11:14.670
that are the eigenvectors that
correspond to each of these

00:11:14.670 --> 00:11:15.020
eigenvalues.

00:11:15.020 --> 00:11:16.460
So let's graph them.

00:11:16.460 --> 00:11:19.750
So if we go to R2, let
me draw my axes, this

00:11:19.750 --> 00:11:22.120
is my vertical axis.

00:11:22.120 --> 00:11:26.230
That's my horizontal axis.

00:11:26.230 --> 00:11:30.430
So all of the vectors that
correspond to lambda equal 5

00:11:30.430 --> 00:11:33.970
are along the line 1/2, 1.

00:11:33.970 --> 00:11:35.460
Or the span of 1/2, 1.

00:11:35.460 --> 00:11:37.320
So that is 1.

00:11:37.320 --> 00:11:38.750
That is 1.

00:11:38.750 --> 00:11:42.170
So you go 1/2 and 1
just like that.

00:11:42.170 --> 00:11:46.150
So that's that vector,
spanning vector.

00:11:46.150 --> 00:11:49.770
But anything along the span of
this, all the multiples of

00:11:49.770 --> 00:11:52.060
this, are going to be
valid eigenvectors.

00:11:52.060 --> 00:11:55.360
So anything along that line,
all of the vectors when you

00:11:55.360 --> 00:11:56.920
draw them in standard
position, point to a

00:11:56.920 --> 00:11:59.540
point on that line.

00:11:59.540 --> 00:12:01.900
All of these vectors, any vector
on there is going to be

00:12:01.900 --> 00:12:05.430
a valid eigenvector and the
corresponding eigenvalue is

00:12:05.430 --> 00:12:07.020
going to be equal to 5.

00:12:07.020 --> 00:12:11.250
So you give me this
guy right here.

00:12:11.250 --> 00:12:13.270
When you apply the
transformation, it's going to

00:12:13.270 --> 00:12:14.805
be five times this guy.

00:12:18.120 --> 00:12:21.040
If this guy is x, t of
x is going to be

00:12:21.040 --> 00:12:22.870
five times this guy.

00:12:22.870 --> 00:12:25.940
Whatever vector you give along
this line, the transformation

00:12:25.940 --> 00:12:27.860
of that guy, the transformation
is literally,

00:12:27.860 --> 00:12:31.140
multiplying it by
the matrix A.

00:12:31.140 --> 00:12:32.330
Where did I have the matrix A?

00:12:32.330 --> 00:12:34.040
The matrix A right up there.

00:12:34.040 --> 00:12:36.660
You're essentially just scaling
this guy by 5 in

00:12:36.660 --> 00:12:37.320
either direction.

00:12:37.320 --> 00:12:40.160
This is for lambda equal 5.

00:12:40.160 --> 00:12:44.390
And for lambda equals 1, it's
the span of this vector, which

00:12:44.390 --> 00:12:48.730
is minus 1, 1.

00:12:48.730 --> 00:12:51.100
Which looks like this.

00:12:51.100 --> 00:12:53.460
So this vector looks
like that.

00:12:53.460 --> 00:12:54.800
We care about the span of it.

00:12:58.170 --> 00:13:01.370
Any vector that when you draw in
standard position lies, or

00:13:01.370 --> 00:13:05.770
points to, points on this line,
will be an eigenvector

00:13:05.770 --> 00:13:08.170
for the eigenvalue minus 1.

00:13:08.170 --> 00:13:11.500
So lambda equals minus 1.

00:13:11.500 --> 00:13:13.860
Let's say you take the
spanning vector here.

00:13:13.860 --> 00:13:16.010
You apply the transformation,
you're going to get

00:13:16.010 --> 00:13:17.730
minus 1 times it.

00:13:17.730 --> 00:13:20.240
So if this is x, the
transformation of x is going

00:13:20.240 --> 00:13:21.100
to be that right there.

00:13:21.100 --> 00:13:23.670
Same length, just in the
opposite direction.

00:13:23.670 --> 00:13:26.340
If you have this guy right
here, you apply the

00:13:26.340 --> 00:13:29.510
transformation, it's going to
be in the same spanning line

00:13:29.510 --> 00:13:30.570
just like that.

00:13:30.570 --> 00:13:35.580
So the two eigenspaces for the
matrix-- where did I write it?

00:13:35.580 --> 00:13:40.610
I think it was the matrix
1, 2, 3-- 1, 2, 4, 3.

00:13:40.610 --> 00:13:45.280
The two eigenvalues were
5 and minus 1.

00:13:45.280 --> 00:13:48.220
And then it has an infinite
number of eigenvectors, so

00:13:48.220 --> 00:13:50.890
they actually create
two eigenspaces.

00:13:50.890 --> 00:13:54.040
Each of them correspond to
one of the eigenvalues.

00:13:54.040 --> 00:13:57.270
And these lines represent
those two eigenspaces.

00:13:57.270 --> 00:13:59.480
You give me any vector in
either of these sets and

00:13:59.480 --> 00:14:02.360
they're going to be
an eigenvector.

00:14:02.360 --> 00:14:03.520
I'm using the word
vector too much.

00:14:03.520 --> 00:14:05.590
You give me any vector in either
of these sets, and they

00:14:05.590 --> 00:14:08.420
will be an eigenvector
for our matrix A.

00:14:08.420 --> 00:14:10.520
And then, depending on which
line it is, we know what their

00:14:10.520 --> 00:14:11.810
transformation is going to be.

00:14:11.810 --> 00:14:13.930
If it's going to be on
this guy, we take the

00:14:13.930 --> 00:14:16.060
transformation, the resulting
vector's going to be five

00:14:16.060 --> 00:14:17.620
times the vector.

00:14:17.620 --> 00:14:20.510
If you take one of these
eigenvectors and you transform

00:14:20.510 --> 00:14:22.920
it, the resulting transformation
of the vector's

00:14:22.920 --> 00:14:25.550
going to be minus 1
times that vector.

00:14:25.550 --> 00:14:28.900
Anyway, we now know what
eigenvalues, eigenvectors,

00:14:28.900 --> 00:14:30.010
eigenspaces are.

00:14:30.010 --> 00:14:33.560
And even better, we know how
to actually find them.


Example solving for the eigenvalues of a 2x2 matrix

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.630
In the last video we were able
to show that any lambda that

00:00:03.630 --> 00:00:09.910
satisfies this equation for some
non-zero vectors, V, then

00:00:09.910 --> 00:00:13.510
the determinant of lambda times
the identity matrix

00:00:13.510 --> 00:00:15.560
minus A, must be equal to 0.

00:00:15.560 --> 00:00:24.620
Or if we could rewrite this as
saying lambda is an eigenvalue

00:00:24.620 --> 00:00:31.780
of A if and only if-- I'll
write it as if-- the

00:00:31.780 --> 00:00:37.250
determinant of lambda times the
identity matrix minus A is

00:00:37.250 --> 00:00:39.860
equal to 0.

00:00:39.860 --> 00:00:42.230
Now, let's see if we can
actually use this in any kind

00:00:42.230 --> 00:00:45.690
of concrete way to figure
out eigenvalues.

00:00:45.690 --> 00:00:48.880
So let's do a simple 2
by 2, let's do an R2.

00:00:48.880 --> 00:00:58.000
Let's say that A is equal to
the matrix 1, 2, and 4, 3.

00:00:58.000 --> 00:01:01.790
And I want to find the
eigenvalues of A.

00:01:01.790 --> 00:01:11.610
So if lambda is an eigenvalue
of A, then this right here

00:01:11.610 --> 00:01:16.330
tells us that the determinant
of lambda times the identity

00:01:16.330 --> 00:01:20.140
matrix, so it's going to be
the identity matrix in R2.

00:01:20.140 --> 00:01:29.230
So lambda times 1, 0, 0, 1,
minus A, 1, 2, 4, 3, is going

00:01:29.230 --> 00:01:30.320
to be equal to 0.

00:01:30.320 --> 00:01:32.510
Well what does this equal to?

00:01:32.510 --> 00:01:36.170
This right here is
the determinant.

00:01:36.170 --> 00:01:39.740
Lambda times this is just lambda
times all of these

00:01:39.740 --> 00:01:42.790
terms. So it's lambda times 1
is lambda, lambda times 0 is

00:01:42.790 --> 00:01:47.260
0, lambda times 0 is 0, lambda
times 1 is lambda.

00:01:47.260 --> 00:01:49.910
And from that we'll
subtract A.

00:01:49.910 --> 00:01:56.130
So you get 1, 2, 4, 3, and
this has got to equal 0.

00:01:56.130 --> 00:01:58.820
And then this matrix, or this
difference of matrices, this

00:01:58.820 --> 00:02:00.580
is just to keep the
determinant.

00:02:00.580 --> 00:02:03.360
This is the determinant of.

00:02:03.360 --> 00:02:06.670
This first term's going
to be lambda minus 1.

00:02:06.670 --> 00:02:11.540
The second term is 0 minus
2, so it's just minus 2.

00:02:11.540 --> 00:02:15.570
The third term is 0 minus
4, so it's just minus 4.

00:02:15.570 --> 00:02:18.310
And then the fourth term
is lambda minus

00:02:18.310 --> 00:02:23.310
3, just like that.

00:02:23.310 --> 00:02:25.620
So kind of a shortcut to
see what happened.

00:02:25.620 --> 00:02:29.560
The terms along the diagonal,
well everything became a

00:02:29.560 --> 00:02:30.550
negative, right?

00:02:30.550 --> 00:02:31.700
We negated everything.

00:02:31.700 --> 00:02:33.350
And then the terms around
the diagonal, we've got

00:02:33.350 --> 00:02:34.260
a lambda out front.

00:02:34.260 --> 00:02:37.930
That was essentially the
byproduct of this expression

00:02:37.930 --> 00:02:38.900
right there.

00:02:38.900 --> 00:02:41.990
So what's the determinant
of this 2 by 2 matrix?

00:02:41.990 --> 00:02:45.950
Well the determinant of this is
just this times that, minus

00:02:45.950 --> 00:02:46.940
this times that.

00:02:46.940 --> 00:02:58.260
So it's lambda minus 1, times
lambda minus 3, minus these

00:02:58.260 --> 00:03:00.170
two guys multiplied
by each other.

00:03:00.170 --> 00:03:04.480
So minus 2 times minus 4
is plus eight, minus 8.

00:03:04.480 --> 00:03:09.470
This is the determinant of this
matrix right here or this

00:03:09.470 --> 00:03:12.920
matrix right here, which
simplified to that matrix.

00:03:12.920 --> 00:03:17.510
And this has got to
be equal to 0.

00:03:17.510 --> 00:03:20.180
And the whole reason why that's
got to be equal to 0 is

00:03:20.180 --> 00:03:22.810
because we saw earlier,
this matrix has a

00:03:22.810 --> 00:03:24.615
non-trivial null space.

00:03:24.615 --> 00:03:27.860
And because it has a non-trivial
null space, it

00:03:27.860 --> 00:03:29.790
can't be invertible and
its determinant has

00:03:29.790 --> 00:03:31.530
to be equal to 0.

00:03:31.530 --> 00:03:33.160
So now we have an interesting
polynomial

00:03:33.160 --> 00:03:33.880
equation right here.

00:03:33.880 --> 00:03:36.030
We can multiply it out.

00:03:36.030 --> 00:03:37.030
We get what?

00:03:37.030 --> 00:03:37.960
Let's multiply it out.

00:03:37.960 --> 00:03:46.280
We get lambda squared, right,
minus 3 lambda, minus lambda,

00:03:46.280 --> 00:03:50.880
plus 3, minus 8,
is equal to 0.

00:03:50.880 --> 00:04:00.330
Or lambda squared, minus
4 lambda, minus

00:04:00.330 --> 00:04:04.710
5, is equal to 0.

00:04:04.710 --> 00:04:09.600
And just in case you want to
know some terminology, this

00:04:09.600 --> 00:04:12.520
expression right here is known
as the characteristic

00:04:12.520 --> 00:04:13.770
polynomial.

00:04:19.100 --> 00:04:21.860
Just a little terminology,
polynomial.

00:04:21.860 --> 00:04:24.430
But if we want to find the
eigenvalues for A, we just

00:04:24.430 --> 00:04:25.775
have to solve this right here.

00:04:25.775 --> 00:04:28.310
This is just a basic
quadratic problem.

00:04:28.310 --> 00:04:29.600
And this is actually
factorable.

00:04:29.600 --> 00:04:32.180
Let's see, two numbers and you
take the product is minus 5,

00:04:32.180 --> 00:04:34.250
when you add them
you get minus 4.

00:04:34.250 --> 00:04:39.760
It's minus 5 and plus 1, so you
get lambda minus 5, times

00:04:39.760 --> 00:04:42.580
lambda plus 1, is equal
to 0, right?

00:04:42.580 --> 00:04:47.190
Minus 5 times 1 is minus 5, and
then minus 5 lambda plus 1

00:04:47.190 --> 00:04:50.260
lambda is equal to
minus 4 lambda.

00:04:50.260 --> 00:04:52.970
So the two solutions of our
characteristic equation being

00:04:52.970 --> 00:04:56.740
set to 0, our characteristic
polynomial, are lambda is

00:04:56.740 --> 00:05:02.090
equal to 5 or lambda is
equal to minus 1.

00:05:02.090 --> 00:05:05.240
So just like that, using the
information that we proved to

00:05:05.240 --> 00:05:07.970
ourselves in the last video,
we're able to figure out that

00:05:07.970 --> 00:05:15.610
the two eigenvalues of A are
lambda equals 5 and lambda

00:05:15.610 --> 00:05:17.320
equals negative 1.

00:05:17.320 --> 00:05:19.500
Now that only just solves part
of the problem, right?

00:05:19.500 --> 00:05:22.570
We know we're looking
for eigenvalues and

00:05:22.570 --> 00:05:24.800
eigenvectors, right?

00:05:24.800 --> 00:05:28.660
We know that this equation can
be satisfied with the lambdas

00:05:28.660 --> 00:05:30.700
equaling 5 or minus 1.

00:05:30.700 --> 00:05:33.630
So we know the eigenvalues, but
we've yet to determine the

00:05:33.630 --> 00:05:35.610
actual eigenvectors.

00:05:35.610 --> 00:05:37.660
So that's what we're going
to do in the next video.


Proof of formula for determining eigenvalues

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:06.100
I've got a transformation, m
that's a mapping from Rn to

00:00:06.100 --> 00:00:10.420
Rn, and it can be represented
by the matrix A.

00:00:10.420 --> 00:00:14.310
So the transformation of x
is equal to A times x.

00:00:14.310 --> 00:00:17.500
We saw in the last video it's
interesting to find the

00:00:17.500 --> 00:00:20.480
vectors that only get scaled
up or down by the

00:00:20.480 --> 00:00:21.300
transformation.

00:00:21.300 --> 00:00:23.910
So we're interested in the
vectors where I take the

00:00:23.910 --> 00:00:27.220
transformation of some
special vector v.

00:00:27.220 --> 00:00:29.580
It equals of course,
A times v.

00:00:29.580 --> 00:00:32.250
And we say it only gets scaled
up by some factor,

00:00:32.250 --> 00:00:34.160
lambda times v.

00:00:34.160 --> 00:00:35.940
And these are interesting
because they make for

00:00:35.940 --> 00:00:38.180
interesting basis vectors.

00:00:38.180 --> 00:00:40.580
You know, the transformation
matrix in the alternate

00:00:40.580 --> 00:00:42.680
basis-- this is one of
the basis vectors.

00:00:42.680 --> 00:00:44.220
It might be easier to compute.

00:00:44.220 --> 00:00:46.340
Might make for good coordinate
systems. But they're in

00:00:46.340 --> 00:00:47.700
general, interesting.

00:00:47.700 --> 00:00:51.660
And we call vectors v that
satisfy this, we call them

00:00:51.660 --> 00:00:52.910
eigenvectors.

00:00:55.130 --> 00:01:01.560
And we call their scaling
factors the eigenvalues

00:01:01.560 --> 00:01:04.610
associated with this
transformation and that

00:01:04.610 --> 00:01:06.350
eigenvector.

00:01:06.350 --> 00:01:08.230
You know, hopefully from that
last video, we have a little

00:01:08.230 --> 00:01:09.890
bit of appreciation of
why they're useful.

00:01:09.890 --> 00:01:12.460
But now in this video let's at
least try to determine what

00:01:12.460 --> 00:01:13.200
some of them are.

00:01:13.200 --> 00:01:15.260
You know, based on what we know
so far, if you show me an

00:01:15.260 --> 00:01:18.930
eigenvector I can verify that it
definitely is the case, or

00:01:18.930 --> 00:01:19.620
an eigenvalue.

00:01:19.620 --> 00:01:20.970
I could verify the case.

00:01:20.970 --> 00:01:23.620
But I don't know a systematic
way of solving

00:01:23.620 --> 00:01:24.400
for either of them.

00:01:24.400 --> 00:01:26.750
So let's see if we can come
up with something.

00:01:26.750 --> 00:01:29.840
So in general, we're looking for
solutions to the equation

00:01:29.840 --> 00:01:36.350
A times v is equal
to lambda v.

00:01:36.350 --> 00:01:39.750
It's equal to lambda
times the vector.

00:01:39.750 --> 00:01:42.750
Now one solution might
immediately pop out at you,

00:01:42.750 --> 00:01:46.800
and that's just v is equal
to the 0 vector.

00:01:46.800 --> 00:01:50.210
And that definitely is a
solution, although it's not

00:01:50.210 --> 00:01:54.100
normally considered to be an
eigenvector just because one,

00:01:54.100 --> 00:01:55.610
it's not a useful
basis vector.

00:01:55.610 --> 00:01:57.230
It doesn't add anything
to a basis.

00:01:57.230 --> 00:01:59.800
It doesn't add really the amount
of vectors that you can

00:01:59.800 --> 00:02:02.140
span when you throw the
basis vector in there.

00:02:02.140 --> 00:02:05.420
And also, it's not clear what
is your eigenvalue that's

00:02:05.420 --> 00:02:06.120
associated with it.

00:02:06.120 --> 00:02:08.710
Because if v is equal
to 0, any eigenvalue

00:02:08.710 --> 00:02:09.500
will work for that.

00:02:09.500 --> 00:02:12.320
So normally when we're looking
for eigenvectors, we start

00:02:12.320 --> 00:02:14.955
with the assumption that we're
looking for non-zero vectors.

00:02:14.955 --> 00:02:17.880
So we're looking for vectors
that are not

00:02:17.880 --> 00:02:19.545
equal to the 0 vector.

00:02:19.545 --> 00:02:21.840
So given that, let's see if we
can play around with this

00:02:21.840 --> 00:02:24.660
equation a little bit and see
if we can at least come up

00:02:24.660 --> 00:02:27.260
with eigenvalues maybe
in this video.

00:02:27.260 --> 00:02:31.730
So we subtract Av from both
sides, we get the 0 vector is

00:02:31.730 --> 00:02:38.480
equal to lambda v
minus A times v.

00:02:38.480 --> 00:02:44.460
Now, we can rewrite v as-- v is
just the same thing as the

00:02:44.460 --> 00:02:48.690
identity matrix times v, right?
v is a member of Rn.

00:02:48.690 --> 00:02:50.200
The identity matrix n by n.

00:02:50.200 --> 00:02:52.580
You just multiply and we're
just going to get v again.

00:02:52.580 --> 00:02:55.640
So if I rewrite v this way, at
least on this part of the

00:02:55.640 --> 00:02:58.230
expression-- and let me swap
sides-- so then I'll get

00:02:58.230 --> 00:03:01.390
lambda times-- instead of v
I'll write the identity

00:03:01.390 --> 00:03:08.890
matrix, the n by n identity
matrix times v minus A times v

00:03:08.890 --> 00:03:12.390
is equal to the 0 vector.

00:03:12.390 --> 00:03:17.020
Now I have one matrix times v
minus another matrix times v.

00:03:17.020 --> 00:03:19.140
Matrix vector products,
they have the

00:03:19.140 --> 00:03:20.840
distributive property.

00:03:20.840 --> 00:03:26.300
So this is equivalent to the
matrix lambda times the

00:03:26.300 --> 00:03:31.630
identity matrix minus A
times the vector v.

00:03:31.630 --> 00:03:33.810
And that's going to be
equal to 0, right?

00:03:33.810 --> 00:03:35.930
This is just some matrix
right here.

00:03:38.580 --> 00:03:41.180
And the whole reason why I made
this substitution is so I

00:03:41.180 --> 00:03:44.770
can write this as a matrix
vector product instead of just

00:03:44.770 --> 00:03:47.000
a scalar vector product.

00:03:47.000 --> 00:03:49.340
And that way I was able to
essentially factor out the v

00:03:49.340 --> 00:03:52.700
and just write this whole
equation as essentially, some

00:03:52.700 --> 00:03:56.350
matrix vector product
is equal to 0.

00:03:56.350 --> 00:04:00.760
Now, in order-- if we assume
that this is the case, and

00:04:00.760 --> 00:04:04.030
we're assuming-- remember,
we're assuming that v

00:04:04.030 --> 00:04:05.920
does not equal 0.

00:04:05.920 --> 00:04:07.890
So what does this mean?

00:04:07.890 --> 00:04:13.980
So we know that v is a member
of the null space of this

00:04:13.980 --> 00:04:15.990
matrix right here.

00:04:15.990 --> 00:04:19.339
Let me write this down. v is a
member of the null space of

00:04:19.339 --> 00:04:23.380
lambda I sub n minus A.

00:04:23.380 --> 00:04:25.040
I know that might look a little
convoluted to you right

00:04:25.040 --> 00:04:28.890
now, but just imagine this
is just some matrix B.

00:04:28.890 --> 00:04:29.840
It might make it simpler.

00:04:29.840 --> 00:04:31.580
This is just some matrix
here, right?

00:04:31.580 --> 00:04:32.470
That's B.

00:04:32.470 --> 00:04:33.750
Let's make that substitution.

00:04:33.750 --> 00:04:38.460
Then this equation just becomes
Bv is equal to 0.

00:04:38.460 --> 00:04:41.190
Now, if we want to look at the
null space of this, the null

00:04:41.190 --> 00:04:48.940
space of B is all of the vectors
x that are a member of

00:04:48.940 --> 00:04:55.450
Rn such that B times
x is equal to 0.

00:04:55.450 --> 00:04:59.090
Well, v is clearly one
of those guys, right?

00:04:59.090 --> 00:05:01.560
Because B times v
is equal to 0.

00:05:01.560 --> 00:05:04.560
We're assuming B solves this
equation and that gets all the

00:05:04.560 --> 00:05:07.940
way to the assumption that B
must solve this equation.

00:05:07.940 --> 00:05:10.060
And v is not equal to 0.

00:05:10.060 --> 00:05:13.320
So v is a member of the null
space and this is a nontrivial

00:05:13.320 --> 00:05:14.180
member of the null space.

00:05:14.180 --> 00:05:17.030
We already said the 0 vector is
always going to be a member

00:05:17.030 --> 00:05:19.200
of the null space, and it
would make this true.

00:05:19.200 --> 00:05:20.630
But we're assuming
v is non-zero.

00:05:20.630 --> 00:05:23.170
We're only interested in
non-zero eigenvectors.

00:05:23.170 --> 00:05:27.000
And that means that this guy's
null space has to be

00:05:27.000 --> 00:05:29.070
nontrivial.

00:05:29.070 --> 00:05:36.100
So this means that the null
space of lambda In minus A is

00:05:36.100 --> 00:05:37.350
nontrivial.

00:05:40.240 --> 00:05:43.360
The 0 vector is not
the only member.

00:05:43.360 --> 00:05:49.650
And you might remember before,
that the only time-- let me

00:05:49.650 --> 00:05:50.650
write this in general.

00:05:50.650 --> 00:05:52.170
If I have some matrix--
I don't know.

00:05:52.170 --> 00:05:53.270
I've used A and B.

00:05:53.270 --> 00:05:55.235
Let's say I have
some matrix D.

00:05:59.050 --> 00:06:08.310
D's columns are linearly
independent if and only if the

00:06:08.310 --> 00:06:18.390
null space of D only contains
the 0 vector.

00:06:18.390 --> 00:06:19.030
Right?

00:06:19.030 --> 00:06:23.350
So if we have some matrix here
whose null space does not only

00:06:23.350 --> 00:06:25.130
contain the 0 vector,
then it has

00:06:25.130 --> 00:06:26.670
linearly dependent columns.

00:06:31.180 --> 00:06:33.830
And I just wrote that there to
kind of show you what we do

00:06:33.830 --> 00:06:37.110
know and the fact that this
one doesn't have a trivial

00:06:37.110 --> 00:06:39.220
null space tells us that we're
dealing with linearly

00:06:39.220 --> 00:06:40.670
dependent columns.

00:06:40.670 --> 00:06:44.750
So lambda In minus A-- it looks
all fancy, but this is

00:06:44.750 --> 00:06:52.590
just a matrix-- must have
linearly dependent columns.

00:06:52.590 --> 00:06:56.120
Or another way to say that is,
if you have linearly dependent

00:06:56.120 --> 00:07:02.220
columns, you're not invertible,
which also means

00:07:02.220 --> 00:07:04.990
that your determinate
must be equal to 0.

00:07:04.990 --> 00:07:05.740
All of these are true.

00:07:05.740 --> 00:07:08.000
If your determinate is equal to
0, you're not going to be

00:07:08.000 --> 00:07:09.230
invertible.

00:07:09.230 --> 00:07:10.770
You're going to have linearly
dependent columns.

00:07:10.770 --> 00:07:13.000
If your determinate is equal
to 0, then that also means

00:07:13.000 --> 00:07:17.850
that you have nontrivial members
in your null space.

00:07:17.850 --> 00:07:20.460
And so, if your determinate is
equal to 0 that means there's

00:07:20.460 --> 00:07:27.030
some lambdas for which this is
true, for non-zero vectors v.

00:07:27.030 --> 00:07:32.530
So, if there are some solutions,
if there are some

00:07:32.530 --> 00:07:39.160
non-zero vector v's that satisfy
this equation, then

00:07:39.160 --> 00:07:46.220
this matrix right here must
have a determinate of 0.

00:07:46.220 --> 00:07:46.990
And it goes the other way.

00:07:46.990 --> 00:07:50.830
If this guy has a determinate of
0, then there must be-- or

00:07:50.830 --> 00:07:52.890
if there's some the lambdas
that make this guy have a

00:07:52.890 --> 00:07:56.320
determinate of 0, then those
lambdas are going to satisfy

00:07:56.320 --> 00:07:57.072
this equation.

00:07:57.072 --> 00:07:57.970
And you could go
the other way.

00:07:57.970 --> 00:07:59.640
If there's some lambdas that
satisfy this, then those

00:07:59.640 --> 00:08:03.060
lambdas are going to make this
matrix have a 0 determinate.

00:08:07.030 --> 00:08:08.280
Let me write this.

00:08:10.890 --> 00:08:29.620
Av is equal to lambda v for
non-zero v's if and only if

00:08:29.620 --> 00:08:37.409
the determinate of lambda
In minus A is

00:08:37.409 --> 00:08:41.080
equal to the 0 vector.

00:08:41.080 --> 00:08:41.860
No, not the 0 vector.

00:08:41.860 --> 00:08:43.159
Sorry, it's just equal to 0.

00:08:43.159 --> 00:08:45.280
The determinate is just
a scalar factor.

00:08:45.280 --> 00:08:46.590
And so that's our
big takeaway.

00:08:46.590 --> 00:08:47.830
And I know what you're saying
now, how is that

00:08:47.830 --> 00:08:49.270
useful for me, Sal?

00:08:49.270 --> 00:08:51.040
You know, we did all of
this manipulation.

00:08:51.040 --> 00:08:53.100
I talked a little bit about
the null spaces.

00:08:53.100 --> 00:08:57.000
And my big takeaway is, is that
in order for this to be

00:08:57.000 --> 00:09:00.690
true for some non-zero vectors
v, then lambda

00:09:00.690 --> 00:09:02.085
has to be some value.

00:09:02.085 --> 00:09:05.090
So if I take the determinate of
lambda times the identity

00:09:05.090 --> 00:09:09.050
matrix minus A, it has
got to be equal to 0.

00:09:09.050 --> 00:09:10.670
And the reason why this is
useful is that you can

00:09:10.670 --> 00:09:14.820
actually set this equation up
for your matrices, and then

00:09:14.820 --> 00:09:15.650
solve for your lambdas.

00:09:15.650 --> 00:09:18.460
And we're going to do that
in the next video.