Singularity of symmetric block matrix with singular diagonal blocks

by Minji Kim   Last Updated September 11, 2019 19:20 PM

I proved that following statement holds true and I think I've seen this somewhere before, but I cannot find the reference that explicitly states about this:

$\begin{bmatrix}A & B \\ B^T & 0 \end{bmatrix}$ is invertible if and only if $C^TAC$ is invertible where $A \in \Re^{n \times n}$ is a symmetric matrix, $B \in \Re^{n \times p}$ is a full column rank matrix ($n>p$), and $C \in \Re^{n \times (n-p)}$ is a full column rank matrix with $C^TB = 0$.

How can I find a reference that states exactly about this statement?

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