Permutation matrix using block partitioned matrix

Question

I am given the following theorem: Let $X\in\mathbf{K}^{n\times n}$ be a block partitioned matrix $X=\left[\begin{array}{c|c}A & B \\ \hline C& D \end{array} \right]$ with $A\in\mathbf{K}^{k\times k}$ and $D\in\mathbf{K}^{\ell\times \ell}$, then $\Pi^{-1} X\Pi=\left[\begin{array}{c|c}D & C \\ \hline B& A \end{array} \right]$ using $\Pi=\left[\begin{array}{c|c}0 & {\rm Id}_{\ell} \\ \hline {\rm Id}_{k}& 0 \end{array} \right]$. Of course this is easy to check given that $\Pi^{-1}=\Pi^\top$.

My book gives the following example of a similarity transformation using a $\Pi$ given a block partitioned matrix that consists of 9 blocks:

I am trying to understand what happens here, although I am not able to generalize the result from the theorem into the $3\times 3$-case. Could someone help clarify?

Answer 1

To do this problem you'd want an additional theorem. Keep $X$, $A$, etc. as above, but compute $\Pi^{-1}X\Pi$ for a permutation matrix of the form \begin{equation} \Pi = \left[\begin{array}{c|c} \Pi_k & 0\\ \hline 0&\mathrm{Id}_l\end{array}\right], \end{equation} where $\Pi_k$ is a $k \times k$ permutation matrix. You can use the formula you obtain as the first step in doing the stated problem. In the next step you can apply the theorem you were given to the $5 \times 5$ matrix $\Pi_k$ to finish the problem.

Answer 2

The matrix $\Pi$ is a permutation matrix whose last column is $e_6$. Thus, this column has no effect in the product of $\Pi^{-1}A\Pi$. Hence, the action of $\Pi$ is concentrated on the upper left corner and it is of the form $\begin{bmatrix} 0& I_3\\I_2& 0\end{bmatrix}$. Now you can use the block 2-by-2 case applied to the first 5 rows and first 5 columns of $A$ by applying what you learned in the block 2-by-2 case.