1. ## General Questions about Eigenvalues and eigenvectors

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 $\displaystyle A^t$ have the same eigenvalues? Do A and $\displaystyle A^t$ have the same eigenvectors? If not can someone please give an example?

Thanks!

2. For 1 & 2;

if A & B are similar matrices they have same characteristic polynomial so...

use my above argument to prove q. 2.

3. For some reason I can't relate eigenvectors to characteristic polynomials.

4. You might not be able to. It is possible to have two matrices with the same eigenvalues but different eigenvectors.

Try to find an example for yourself.

5. The sticky about proofs will show examples.

6. I'm afraid AlsoSprachZarathustra misunderstood when you said "(I know they have the same eigenvalues...)".

If A and B are similar matrices then $\displaystyle B= P^{-1}AP$ for some invertible P. If x is an eigenvector of A, with eigenvalue $\displaystyle \lambda$, let $\displaystyle y= P^{-1}x$. Then $\displaystyle By= P^{-1}APy= P^{-1}APP^{-1}x$$\displaystyle = P^{-1}Ax= P^{-1}\lambda x= \lambda P^{1}x= \lambda y$.

Thus, if similar matrices have the same eigenvalues but NOT, in general, the same eigenvectors.

7. Thus, if similar matrices have the same eigenvalues but NOT, in general, the same eigenvectors.
Does this hold true also for it's $\displaystyle A^t$

8. Originally Posted by jayshizwiz
Does this hold true also for it's $\displaystyle A^t$
If you would go to the sticky in the forum, you would have your answers since I have pdf showing these scenarios and many more available for download.

9. O, I just noticed where the stickies are lol... Thanks.