hello everyone I am TARUN KAUSHIK and today
in this small video i will be discussing
advanced integration and the question is
we have to prove that the eigenvalues of
the hermitian matrix orient ok so first
of all we must know that to be the
Commission metrics and video its
property so I'm taking organs that
solving this question by taking an
example this is a metric savings you buy
these metrics
ok so now for calculating the emission
of metrics we first we need to calculate
the transport of that matrix and after
calculating the transport we need to do
the conjugate of that when the if we do
that transpose transpose matrix
conjugate we get the symmetrix our
original matrix for example if i do the
conjugate of this matrix but I get this-
i get plus I so it becomes to the less I
and this become 3-3 I ok and this
becomes 2 minus I and this becomes 3
plus TI and this become minus of siam
minus 2i and this become plus 2i ok now
after doing the conjugate of this matrix
and if i do the transpose of this matrix
and then again these metrics against
their this minus sign goes to this place
and this plus sign organs a minus sign
goes to this place so it becomes the
again this is the same matrix which is
our a matrix
ok so we can write this a close to a
hermitian so the matrix a is also equal
to its a hard mission
ok and they are measuring nothing you
just that transport of the conjugate or
you can see that contribute to the
transport you can rewrite in any way
ok so in the UN in the UPC you do not
need to do this stuff
okay you can start from this point that
in the property of the mission metrics
is that a equals to its harmonic
considered hermitian matrix addition
property a closer age or reconsider that
that is equals to a start to restyle
means that conjugate ok now
really portion starts from here i am
multiplying this a this movie is I
convector ok and lambda Jagan value then
into eigenvector will be close to lambda
into that I came back I convector ok so
this will use my digital umeda yeah one
equation
ok so a into the eigenvector will be
close to lambda that is i can value
window icon vector is the property
ok very simple you can see that it's
just a combination of the linear
equations which are given an equation
itself it is just only that now the rear
portion starts from this point i have to
multiply into the pre multiplication by
this vector back this vector V so we i'm
multiplying the family planning this
equation by the natural formation of
this vector V on both side so if i do
the vh both side we add to both sides
and what do I get this is a lambda and
lambda is a constant down so i can take
slammed outside so it becomes uvh into V
okay now this Humvee are having at this
point now if we take our mission both
side then what we get
before using that we must use this
property we must aware of this property
a be raised to the power minus one so if
the two mattresses are given and we have
to calculate its inverse so then what
will be close to being worse into
inverse ok so by the same property we
have to used in calculating the
commission so then in the reverse order
we have to take so this insight potion
this is V so we have to write this react
here and we are two tickets
our mission so we catch these entities
and a raise to the power or consider not
rise to develop it's a hatch ketchup is
the hermitian of this this is very edge
and the permission of the hermitian
organs that transport transport Judah
method this at ads get cancer we are
left with this vector
so this is the vector ok because the
transport of the transpose equals to the
vector itself organs that time metric
system may transport and the transport
of that you should review mean that only
organ CA inverse a day to the power
inverse so that inverse inverse get
cancer so do you mean that a value only
in the same way if i do the hermitian to
time should we give me the to reconsider
to the vector itself that is V self ok
and the lambda chi concert is why you
get so i take it outside should become a
harem permission and this V again this
recognition and this week our mission
into her mission get cancer so we are
left with only V okay
vector we now from the equation 1 the
very origin one are the equation 1
states that this a which is not engage
the hermitian of the metrics or itself
so this eh
we can't replace this is by a only by
using this property
ok so this and this can be you can see
that replaced so then I put the value of
a hermitian that is equal to a so I get
the situation
ok now you can see from the equation 2
and 3 visual verification number two
this is our equation number 2 ok we
fetch every riach AV water left hand
side are equal so the right-hand side
should be also equal so this portion
should be close to this portion
ok so i put lambda vh and thought we
equal to lambda star tour going to learn
to contribute we edge and we so from
where this and even that lambda star
comes from this is lambda at lambda that
means our mission our mission is nothing
it's just the transport of the conjugate
or going to contribute transport so
lambda is just a value so we can just to
know that they will be no transport so
will be only conjugate so i take it
contribute to your ok now this is the
conjugate now this term and this term
but a similar to David it will get
cancelled
now we are left with only lambda and
lambda star number should because two
numbers start now a damaging that
language was to a plus IB so it will be
each country will be a minus IB okay
three equals three be so when these two
conditions will be equal when a class 2a
that is only the real personal equal if
if this condition horse then this be
equals to minus B & B plus to minus B
and dismay or from their value becomes
out will be 0 so it means and the value
or be that is 0 of n destroyed because
otherwise the value of a should because
it means the value of the lambdas would
be real working 200 you we are not
taking it to a jeweler can be but in the
real portion disposed this should be
equal only when a and B even that is
equals to a there's no possibility that
Abby is equals to minus B okay so only
at the point of a because you know that
there is only appropriate program there
this because to minus of the value would
you know the only value which is
negative which is equal to the negative
of itself you become 0 is equal to minus
zero but one is not equal to minus 1 ok
so what is not equal to minus 1 so this
property or else you're going to have
hold only when n is equals to a ok ended
this property's only valid when the
values are real only do the proof
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