My question is a really straightforward one: Is there an easier way to find the eigenvalues and/or eigenvectors of a 2×2 block diagonal matrix other than direct diagonalization of the whole matrix?
$ \left( \begin{array}{ccc}
A & 0 \\
0 & B \end{array} \right )$
Here $A$, $B$ are $n \times n$ symmetric matrices.
Best Answer
Since $A$ and $B$ are symmetrical, we can diagonalize them in the way such that $A=UD_AU^T$ and $B=VD_BV^T$.
Hence,
$\left(\begin{array}{cc} A & 0 \\ 0 & B \end{array} \right)=\left(\begin{array}{cc} UD_AU^T & 0 \\ 0 & VD_BV^T \end{array} \right)=\left(\begin{array}{cc} U & 0 \\ 0 & V \end{array} \right)\left(\begin{array}{cc} D_A & 0 \\ 0 & D_B \end{array} \right)\left(\begin{array}{cc} U & 0 \\ 0 & V \end{array} \right)^T.$