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Math Help - Prove if B is similar to A then A is similar to B

  1. #1
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    Prove if B is similar to A then A is similar to B

    I am having trouble with this proof.

    A matrix B is similar to A if there exists a nonsingular matrix P such that

    PAP^-1=B

    The question is:

    Prove if B is similar to A then A is similar to B

    I have:

    Assume B is similar to A. Then there exists P such that

    PAP^-1=B.

    multiplying both sides by P gives

    PA=BP

    then multiplying both sides by inverse of P gives

    A=P^-1 *B*P

    but I need to show A=PBP^-1. and I cannot figure it out. Thanks!!!!!!!
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  2. #2
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    Quote Originally Posted by mulaosmanovicben View Post
    I am having trouble with this proof.

    A matrix B is similar to A if there exists a nonsingular matrix P such that

    PAP^-1=B

    The question is:

    Prove if B is similar to A then A is similar to B

    I have:

    Assume B is similar to A. Then there exists P such that

    PAP^-1=B.

    multiplying both sides by P gives

    PA=BP

    then multiplying both sides by inverse of P gives

    A=P^-1 *B*P

    but I need to show A=PBP^-1. and I cannot figure it out. Thanks!!!!!!!

    No, you do NOT need to prove A=PBP^{-1} , which in general isn't true. What you need is to prove that there exists an invertible matrix Q s.t. A=QBQ^{-1}...well, just choose Q=P^{-1}

    Tonio
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  3. #3
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    Feb 2010
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    Thanks Tonio! I actually skipped class the day the professor described invertible matricies (had to work on some homework that was due that day haha) but this solution sounds elegant and beautiful =D Thanks alot!!!!!!!!!!!!

    EDIT: aah now I see invertible just means AB=BA=I
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