# Milnor - Morse Theory, proof of Morse's lemma

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 $$i,j\geq r$$.

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.