In this segment we will talk about some of
the theorems which are related to eigenvalues
and eigenvectors so one of the first theorems
which people talk about is if [A] is a square
matrix (nxn) but it is upper triangular, lower
triangular or diagonal matrix, then the eigenvalues
of [A] are the diagonal entries of [A] so
if you have an upper triangular matrix or
lower triangular matrix or a diagonal matrix
then you will find out the eigenvalues of
[A] you don t have to go through the process
of finding the determinant and things like
that because the eigenvalues of [A] themselves
are the diagonal entries of the [A] matrix
so lets take an example if somebody gives
you a lower triangular matrix like this [A]=
6, 0, 0, 3, -2, 0, 7, 6, 5 so this is a lower
triangular matrix because anything above the
diagonal is zero so that s a lower triangular
matrix since every element above the diagonal
is zero and in this case the three eigenvalues
which you will get for this particular upper
triangular matrix will lower triangular matrix
will be lambda1= 6, lambda2= -2, lambda3=
5 so from this lower triangular matrix you
have three eigenvalues lambda1= 6, lambda2=
-2, lambda3= 5 and that is the end of this
segment
