eigenvalues of a block matrix

**yyalli** · July 12th 2010, 12:04 PM

Hi all,

I want to find eigenvalues of the following block matrix X:
$ 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

**Jose27** · July 12th 2010, 12:43 PM

Originally Posted by yyalli

Hi all,

I want to find eigenvalues of the following block matrix X:
$ 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 $U$ and $V$ commute, then you could use the thoerem that says: If $AC=CA$ (all blocks are n by n) then if:

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

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

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