CHAPTER 10.10: EIGENVALUES AND
EIGENVECTORS: Theorems of eigenvalues and eigenvectors Part 3 of 6 In
this segment we are talking about some theorems of corresponding to
eigenvalues and eigenvectors so one of
the theorems says as follows that [A] is a nxn
matrix so what that means is that [A] is a square matrix then [A] transpose
has same eigenvalues as [A] so what that basically means is that if you have
a square matrix and you are able to find its eigenvalues then if you take its
transpose then it has the same eigenvalues as the matrix so lets suppose that somebody says that this particular
matrix right here [A]= 2, -3.5, 6, 3.5, 5, 2, 8, 1, 8.5 and if you take this
particular matrix right here and you find its eigenvalues by following the
procedure you will get 3 eigenvalues you’ll get λ1=-1.547, λ2=12.33,
λ3=4.711 so these are 3 eigenvalues which you are going to get for that
particular matrix so now if somebody gave me another matrix [B] like this one
so [B]= 2, 3.5, 8, -3.5, 5, 1, 6, 2, 8.5 and gave me a matrix like this one
if I realize that hey this matrix [B] is simply the transpose of the [A]
matrix as you can see that 2 -3.5 6 is right here 3.5 5 2 is right here 8 1
8.5 is right here so the [B] matrix here is same as the [A] transpose then
the eigenvalues of [B] are the same so λ1=-1.547, so the eigenvalues of
[B] λ2=12.33, λ3=4.711and
that is the end of this segment |