Thread: eigenvalues of a block matrix

1. 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(...).

Thanks,
Jong

2. Originally Posted by yyalli
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(...).

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.