Suppose $A$ is a $k_1times k_2$ matrix with real entries, $k_1<k_2$. 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!