Inverse of a $2 \times 2$ block matrix with a zero diagonal block

Question

I want to compute the inverse of the $2\times2$ block matrix

$$\left(\begin{array}{cc} A & P\\ P^T & 0\end{array}\right),$$

where $A$ is an $n\times n$ matrix and $P$ is an $n \times m$ matrix.

I only come across formulas for $$ \left(\begin{array}{cc} A & B\\ C & D\\ \end{array}\right)^{-1} $$ that involve $D^{-1}$. However, in my case $D=0$, such that $D$ has no inverse and I don't know how to continue.

One of the Related questions may be helpful, this one in particular. — pjs36, Mar 01 '16 at 13:48
This might help https://en.wikipedia.org/wiki/Schur_complement — Wintermute, Mar 01 '16 at 13:51

score 3 · Accepted Answer · answered Mar 01 '16 at 13:58

3

One of the formulas here does not involve $D^{-1}$

answered Mar 01 '16 at 13:58

Wouter

8,117

1

Thanks, I just found it myself as well. Wanted to say just that when I saw your answer. I started searching in papers - guess I should trust Wikipedia more often :p – Eric S. Mar 01 '16 at 14:04

Inverse of a $2 \times 2$ block matrix with a zero diagonal block

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