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Thread: Commuting Matrices

  1. #1
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    Commuting Matrices

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

    Find some $\displaystyle \alpha, \beta \in \mathbb{C}$ such that $\displaystyle (2A - t_AI)B=\alpha A + \beta I.$
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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 $\displaystyle n \times n$ 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 $\displaystyle 2\times2$ matrices $\displaystyle A$ and $\displaystyle B$

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

    In particular, if $\displaystyle AB=BA$ then

    Spoiler:

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


    Incidentally, $\displaystyle (AB-BA)^2=\gamma I$ holds for all $\displaystyle 2\times2$ matrices $\displaystyle A$ and $\displaystyle B$. What is $\displaystyle \gamma$?
    Last edited by halbard; Aug 20th 2010 at 01:12 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 $\displaystyle 2\times2$ matrices $\displaystyle A$ and $\displaystyle B$

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


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