Diagonalize 2x2 block matrix where the blocks are diagonal matrices

Question

Say we have a 2x2 block matrix: \begin{equation} M = \begin{pmatrix} I & A \\ A & I\end{pmatrix}. \end{equation}

where A is a diagonal matrix (all diagonal entries are non-zero). What would be the singular vectors and singular values of $M$?

Answer 1

I just found a solution, so I am answering my own question.

A singular vector matrix is

\begin{equation} \frac{1}{\sqrt2}\begin{pmatrix} I & I \\ -I & I\end{pmatrix} \end{equation}.

Proof:

\begin{equation} \frac{1}{2}\begin{pmatrix} I & I \\ -I & I\end{pmatrix}\begin{pmatrix} I & A \\ A & I\end{pmatrix}\begin{pmatrix} I & -I \\ I & I\end{pmatrix} = \begin{pmatrix} I+A & 0 \\ 0 & I-A\end{pmatrix} \end{equation} which is a diagonal matrix.

The rows and columns may need to be permuted such that the diagonal terms of the diagonal matrix are ordered.