For 1 & 2;
if A & B are similar matrices they have same characteristic polynomial so...
use my above argument to prove q. 2.
Hi. I have some questions regarding eigen-related things.
1) If A & B are similar matrices do they have the same eigenvectors? (I know they have the same eigenvalues...) If not, can someone please give an example?
2) A is a square matrix. Do A and have the same eigenvalues? Do A and have the same eigenvectors? If not can someone please give an example?
I'm afraid AlsoSprachZarathustra misunderstood when you said "(I know they have the same eigenvalues...)".
If A and B are similar matrices then for some invertible P. If x is an eigenvector of A, with eigenvalue , let . Then .
Thus, if similar matrices have the same eigenvalues but NOT, in general, the same eigenvectors.