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Let $$ M = \begin{bmatrix} A & B \\ C & D \end{bmatrix} $$ be a non-singular block matrix where $A\in\mathbb R^{p\times p}$ and $D\in\mathbb R^{q\times q}$. I suspected that if $p< q$ then $A-BDC$ is a non-singular matrix.

I have tried hard to find a counter-example with no luck. So I tried to prove it. This is where I hit several dead-ends.

First, if you left-multiply $\begin{bmatrix}I&-B\\0&I\end{bmatrix}$ and right-multiply $\begin{bmatrix}I&0\\C&I\end{bmatrix}$ to $M$, you'll get $$\begin{bmatrix}A-BDC&B(I-D)\\(I+D)C&D\end{bmatrix}$$ but this wouldn't lead to any conclusion.

I also tried to use a pseudo-inverse approach on $[A\quad B]$ but the results are disappointing. So I would appreciate any hints or counter-examples to put an end to this problem.

Rodrigo de Azevedo
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polfosol
  • 9,245

1 Answers1

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The block matrix $$ M = \left[\!\begin{array}{c|cc}0 & 1 & 0 \\ \hline 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\!\right] $$ is non-singular, yet we get $$ A - BDC = 0 - \begin{bmatrix}1 & 0\end{bmatrix}\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix} \begin{bmatrix}1 \\ 0\end{bmatrix} = 0. $$

Misha Lavrov
  • 142,276