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Math Help - Commuting Matrices

  1. #1
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    Commuting Matrices

    [ \star \star ] Let A,B be any 2 \times 2 matrices with entries from \mathbb{C} and AB=BA. Let t_A and I denote the trace of A and the 2 \times 2 identity matrix respectively.

    Find some \alpha, \beta \in \mathbb{C} such that (2A - t_AI)B=\alpha A + \beta I.
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  2. #2
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    i don't have to mention that i'm not interested in a direct solution. anyway, here is a hint for those who like the problem:

    Spoiler:
    a clever use of Cayley-Hamilton theorem will solve the problem very quickly!


    by the way, the result can be extended to n \times n commuting matrices but the solution to this general case is much harder.
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  3. #3
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    Commuting(?) matrices

    This is a particular consequence of a very well-known result. For any 2\times2 matrices A and B

    AB+BA=t_B A+t_A B+(t_{AB}-t_At_B)I

    In particular, if AB=BA then

    Spoiler:

    2AB-t_AB=\alpha A+\beta I where \alpha=t_B and \beta=t_{AB}-t_At_B.


    Incidentally, (AB-BA)^2=\gamma I holds for all 2\times2 matrices A and B. What is \gamma?
    Last edited by halbard; August 20th 2010 at 02:12 AM.
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  4. #4
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    Quote Originally Posted by halbard View Post
    This is a particular consequence of a very well-known result. For any 2\times2 matrices A and B

    AB+BA=t_B A+t_A B+(t_{AB}-t_At_B)I
    well, assuming that the above holds, your solution would obviously be correct. my solution is different from yours.


    Incidentally, (AB-BA)^2=\gamma I holds for all 2\times2 matrices A and B. What is \gamma?
    \gamma = -\det(AB-BA), by Cayley-Hamilton and the fact that t_{AB-BA}=0.
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