# linear algebra – Eigenvalues of a block matrix with zero diagonal blocks

Suppose $$A$$ is a $$k_1times k_2$$ matrix with real entries, $$k_1. Let $$M$$ be the matrix
$$begin{equation} M:=begin{pmatrix} 0_{k_1} & A\ A^top & 0_{k_2} end{pmatrix}, end{equation}$$
where $$0_k$$ denotes the $$ktimes k$$ zero matrix. I know that if $$lambda$$ is an eigenvalue of $$M$$ then $$lambda^2$$ must be an eigenvalue of $$A^top A$$. Since $$k_2>k_1$$, we can immediately conclude that $$M$$ has at least $$k_2 – k_1$$ zero eigenvalues.

I wish to obtain a generalization of this observation in the following sense. Suppose $$A_{12},A_{13}$$ and $$A_{23}$$ are $$k_1times k_2$$, $$k_1times k_3$$ and $$k_2times k_3$$ dimensional matrices respectively and let
$$begin{equation} M:=begin{pmatrix} 0_{k_1} & A_{12} & A_{13} \ A_{12}^top& 0_{k_2} & A_{23} \ A_{13}^top& A_{23}^top& 0_{k_3} end{pmatrix}. end{equation}$$
My conjecture is that if $$k_3>k_1+k_2$$, then $$M$$ contains at least $$k_3-k_1-k_2$$ zero eigenvalues. I can’t figure out how to prove it – any help/hint is appreciated!

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