# Similarity

• November 26th 2011, 02:31 PM
page929
Similarity
Assume all matrices are square. If A is similar to B, then A^2 is similar to B^2. I need to prove or disprove.

I know that to be similar there needs to exist a nonsingular matrix S such that A = S^(-1)BS.

I want to say that it will be true:
AA = [S^(-1)BS][S^(-1)BS] = S^(-1)BBS
A^2 = S^(-1)B^2S
Therefore, A^2 is similar to B^2

Can someone let me know if I am correct and if not, help me out?

Thanks!
• November 26th 2011, 02:36 PM
Deveno
Re: Similarity
that is correct.
• November 26th 2011, 02:36 PM
Also sprach Zarathustra
Re: Similarity
Quote:

Originally Posted by page929
Assume all matrices are square. If A is similar to B, then A^2 is similar to B^2. I need to prove or disprove.

I know that to be similar there needs to exist a nonsingular matrix S such that A = S^(-1)BS.

I want to say that it will be true:
AA = [S^(-1)BS][S^(-1)BS] = S^(-1)BBS
A^2 = S^(-1)B^2S
Therefore, A^2 is similar to B^2

Can someone let me know if I am correct and if not, help me out?

Thanks!

Correct.

EDIT:

BTW, you can replace 2 in every positive integer $n$, and still $A^n$ remain similar to $B^n$.
• November 28th 2011, 12:38 AM
FernandoRevilla
Re: Similarity
More general if $A,B\in\mathbb{F}^{n\times n}$ ( $\mathbb{F}$ field) are similar matrices and $p\in\mathbb{K}[x]$ , then $p(A)$ and $p(B)$ are also similar matrices.