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Math Help - Matrix Formula

  1. #1
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    Matrix Formula

    I have this:
    A=any square matrix over any field F
    p(X)=any polynomial over F
    P=any invertible matrix over F
    B=inverse(P)AP=P^(-1)AP

    I need to show that p(B)=inverse(P)p(A)P=P^(-1)p(A)P.
    How would I prove this?
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  2. #2
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    No it looks like what I wrote.
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  3. #3
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    Is there anyway to show it more algebraically?

    He gives a hint: First do this in the case p(X)=X^m.
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  4. #4
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    Sorry about that.
    It used the important property of similar matricies.
    (P^{-1}AP)^n=P^{-1}A^nP.
    Let the polynomial F[x] be,
    f(x)=a_0+a_1x+a_2x^2+...+a_nx^n.
    Then,
    f(B)=a_0I+a_1B+a_2B^2+...+a_nB^n
    And,
    f(P^{-1}AP)=a_0I+a_1(P^{-1}AP)+a_2(P^{-1}AP)^2+...+a_n(P^{-1}AP)^n
    Use the property mentioned about,
    f(P^{-1}AP)=a_0(P^{-1}P)+a_1(P^{-1}AP)+a_2(P^{-1}A^2P)+...+a_n(P^{-1}A^nP)
    You can use left-right distributive laws in a matrix,
    f(P^{-1}AP)=P^{-1}(a_0+a_1A+a_2A^2+...+a_nA^n)P=P^{-1}f(A)P
    But,
    f(B)=f(P^{-1}AP)
    Thus,
    f(B)=P^{-1}f(A)P

    (The only mistake with you question is that you said a polynomial over a field. That is not what I assumed).
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