1. ## 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!

2. ## Re: Similarity

that is correct.

3. ## Re: Similarity

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 $\displaystyle n$, and still $\displaystyle A^n$ remain similar to $\displaystyle B^n$.

4. ## Re: Similarity

More general if $\displaystyle A,B\in\mathbb{F}^{n\times n}$ ($\displaystyle \mathbb{F}$ field) are similar matrices and $\displaystyle p\in\mathbb{K}[x]$ , then $\displaystyle p(A)$ and $\displaystyle p(B)$ are also similar matrices.