Thread: 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!

that is correct.

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 , and still remain similar to .

More general if ( field) are similar matrices and , then and are also similar matrices.