I would like to prove that KTT matrix defined below has at least $n-m$ positive and at least $m$ negative eigenvalues.
$$ K:= \begin{bmatrix} H & A^{T} \\ A & 0 \end{bmatrix} $$
where $H \in \Bbb R^{n \times n}$ is symmetric, $A \in \Bbb R^{m \times n}$ full rank, $\operatorname{rank} (A) = m$. Assume $H$ is positive definite , $x^{T} H x > 0$ on the nullspace of $A$, $A x = 0$ where $x \neq 0$.
I also wonder if we can say that null($H$) spans the same vectors as null($A$) since it is on the null($A$)? What is the intuitive meaning of being on the nullspace of a matrix? Any hints would be appreciated! Thanks!