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Thread: eigenvalues of a block matrix

  1. #1
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    Jul 2010
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    eigenvalues of a block matrix

    Hi all,

    I want to find eigenvalues of the following block matrix X:
    $\displaystyle
    X = \left[ \begin{array}{cc}
    2U-V & -V \\
    -V & 2U-V \end{array} \right]
    $,
    where U and V are n-by-n block matrices and so the X is a 2n-by-2n matrix.

    Can I further simplify the equation? I'm looking for solutions like eig(...).

    Thank you for your any advice and comment in advance.

    Thanks,
    Jong
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  2. #2
    Super Member
    Joined
    Apr 2009
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    México
    Posts
    721
    Quote Originally Posted by yyalli View Post
    Hi all,

    I want to find eigenvalues of the following block matrix X:
    $\displaystyle
    X = \left[ \begin{array}{cc}
    2U-V & -V \\
    -V & 2U-V \end{array} \right]
    $,
    where U and V are n-by-n block matrices and so the X is a 2n-by-2n matrix.

    Can I further simplify the equation? I'm looking for solutions like eig(...).

    Thank you for your any advice and comment in advance.

    Thanks,
    Jong
    Well, if you knew that $\displaystyle U$ and $\displaystyle V$ commute, then you could use the thoerem that says: If $\displaystyle AC=CA$ (all blocks are n by n) then if:

    $\displaystyle X= \left( \begin{array}{cc} A & B \\ C & D \end{array} \right) $

    then $\displaystyle \det (X) = \det(AD-BC)$ and hopefully this simplifies things for calculating the eigenvalues.
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