Suppose $M$ is an $2n \times 2n$ block matrix:
$$ M = \begin{pmatrix}
A & B \\
C & D
\end{pmatrix}
$$
where $A,B,C$ and $D$ are square matrices of the same order. Are there operations that block-diagonalizes $M$:
$$M \mapsto \begin{pmatrix}
A & 0 \\
0 & D
\end{pmatrix}
?
$$
What about anti-block diagonalize:
$$M \mapsto \begin{pmatrix}
0 & B \\
C & 0
\end{pmatrix}?
$$
We can assume $n \ge 2$ here. If such operations do not exist for every matrix, do these exist for particular matrices (i.e. symmetric, anti-symmetric, and so on)?
EDIT: Such transformations could be, say, a product $SMS$ or $SM S^{-1}$ that does the job, or something related. In other words, can we find other matrices whose products give the block diagonalized or anti-block diagonalized $M$ (i.e. some sort of "projection")?
Best Answer
Denote by $I$ the $n\times n$ identity matrix and define block matrices $$ \mathbf E_{11} = \begin{pmatrix} I & 0 \\ 0 & 0\end{pmatrix},\ \mathbf E_{12} = \begin{pmatrix} 0 & I \\ 0 & 0\end{pmatrix},\ \mathbf E_{21} = \begin{pmatrix} 0 & 0 \\ I & 0\end{pmatrix},\ \mathbf E_{22} = \begin{pmatrix} 0 & 0 \\ 0 & I\end{pmatrix}. $$ Note that for $M=(\begin{smallmatrix} A & B \\ C & D\end{smallmatrix})$ we have \begin{align} \mathbf E_{11} M \mathbf E_{11} &= \begin{pmatrix} A & 0 \\ 0 & 0\end{pmatrix}, & \mathbf E_{11} M \mathbf E_{22} &= \begin{pmatrix} 0 & B \\ 0 & 0\end{pmatrix}, \\ \mathbf E_{22} M \mathbf E_{11} &= \begin{pmatrix} 0 & 0 \\ C & 0\end{pmatrix}, & \mathbf E_{22} M \mathbf E_{22} &= \begin{pmatrix} 0 & 0 \\ 0 & D\end{pmatrix}. \end{align} Hence, you can construct your projections as \begin{align} \begin{pmatrix} A & 0 \\ 0 & D\end{pmatrix} = \mathbf E_{11} M \mathbf E_{11} + \mathbf E_{22} M \mathbf E_{22}, \\ \begin{pmatrix} 0 & B \\ C & 0\end{pmatrix} = \mathbf E_{11} M \mathbf E_{22} + \mathbf E_{22} M \mathbf E_{11}. \end{align}