Given $A\in\mathbb{R}^{m\times n}(m\le n)$ with $rank(A)=m$, let $Z$ be the matrix whose row number is $n$ and the column vectors are any basis of solution space of $A$.
Let $G\in\mathbb{R}^{n\times n}$ be postive-definite.
Prove that the positive/negative index of inertia of $ \left( \begin{matrix} G & A^T \\ A & 0 \end{matrix} \right) = $ the positive/negative index of inertia of $Z^TGZ + m$.