Milnor - Morse Theory, proof of Morse's lemma

Question

Lemma 2.2. Lemma of Morse - Milnor's Morse Theory, application of inverse function theorem.

I have a question about the linked one. I was reading the book "Morse Theory" of Milnor, and I got stuck in the part which is Question 1 in the link. There is a comment below written as:

for Q1: $f$ is supposed to be non-degenerate, so its Hessian matrix has full rank in a nbhd of the critical point. If the $i,j≥r$ part of the Hessian vanished, the crit. pt. would be degenerate. So there is some non-zeroness in that part of the Hessian, and a linear transformation can move that non-zeroness to $H_{r,r}$.

I've understood this comment until "there is some non-zeroness in that part of the Hessian", but I can't see how to make a linear coordinate change to move the nonzero-ness to $H_{r,r}$.

Edit: I actually also cannot see why $H_{i,j}(0)$ is nonzero for some $i,j\geq r$.

Answer 1

The matrix $M=\{H_{i,j}(0)\}_{r\le i,j\le n}$ being symmetric and nondegenerate means it defines an indefinite inner product on $\mathbb{R}^{n-r+1}$. Hence there exists some $y\in \mathbb{R}^{n-r+1}$ such that \begin{align}y^tMy=\pm 1 \end{align} Hence a change of basis from $\{e_r,\dots, e_n\}$ to $\{y, \tilde{e}_{r+1}, \dots, \tilde{e}_n\}$ guarantees that the new $H_{r,r}(0)=\pm 1$. Here $\tilde{e}_i$ denotes some vectors in the subspace generated by $\{e_r,\dots, e_n\}$ such that $\{y, \tilde{e}_{r+1}, \dots, \tilde{e}_n\}$ indeed denotes a basis for this subspace.