A = 000
Find Eigenvalues, its corresponding eigenvectors, and find a matrix W such that W^t*AW = D, where D is a diagnol matrix.(note that W^t represents the transpose of W)
Eigenvalues, Eigenvectors, diagnolization[/B]
The Attempt at a Solution
Question 1 is to find eigenvalues. Since it is already a lower triangular matrix this was easy and I believe the eigenvalues are 1 and 0. The characteristic equation I got was (0-y)(1-y)(1-y).
Question 2 was to find the corresponding eigenvectors. if y1 = 1, then (A-1I) = -100
From that you get x1 = 0. x2 and x3 are free. So a basis for the eigenspace is (v1, v2) where
v1 = 0
v2 = 0
For y = 0, A -0 is just A. you get x2 = 0 and x1 = -x3 with x3 free. Therefore the corresponding eigenvector is
v = -1
Correct me if I'm wrong but I believe up to here I have done everything correct.
The final part of the question confuses me quite a bit . "find a matrix W such that W^t*AW = D, where D is a diagnol matrix."
I thought diagnolization is only possible if a matrix is invertible. Since A has 0 as an eigenvalue, it is not invertible. Did I get the eigenvectors wrong or is D not the diagnolization of A?
Edit: I cannot find a way to get the space formatting correct. Does anyone know how?