# General Questions about Eigenvalues and eigenvectors

• Jun 26th 2010, 10:36 AM
jayshizwiz
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!
• Jun 26th 2010, 10:47 AM
Also sprach Zarathustra
For 1 & 2;

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

use my above argument to prove q. 2.
• Jun 26th 2010, 10:50 AM
jayshizwiz
For some reason I can't relate eigenvectors to characteristic polynomials.
• Jun 26th 2010, 11:25 AM
roninpro
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.
• Jun 26th 2010, 12:49 PM
dwsmith
The sticky about proofs will show examples.
• Jun 26th 2010, 02:02 PM
HallsofIvy
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.
• Jun 26th 2010, 09:16 PM
jayshizwiz
Quote:

Thus, if similar matrices have the same eigenvalues but NOT, in general, the same eigenvectors.
Does this hold true also for it's $\displaystyle A^t$
• Jun 26th 2010, 09:20 PM
dwsmith
Quote:

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.
• Jun 26th 2010, 09:30 PM
jayshizwiz
O, I just noticed where the stickies are lol... Thanks.