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Math Help - Similarity

  1. #1
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    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!
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  2. #2
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    Re: Similarity

    that is correct.
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Similarity

    Quote Originally Posted by page929 View Post
    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.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    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.
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