Just to provide some context, I'm reading a proof of Morse's Lemma in a book called Topology and Geometry for Physicists, and it's not too difficult of a proof, but I don't understand one tiny part. I'm goona try to phrase things in a way so that even if you don't know Morse's lemma, you may be able to see why the following statement is true (because I don't).
Suppose $f: M \to \mathbb{R}$ is smooth, $0$ is a non degenerate critical point and we choose coordinates $(x_1, ... x_n)$ so that
$$\frac{\partial^2 f}{\partial x^2_1}(0) \neq 0 \ \ \ \ {and} \ \ \ \ \ \ f(x) = \sum_{i =1}^n\sum_{j=1}^n{x_i x_j} h_{ij}$$
where $h_{11}(0) = \frac{\partial^2 f}{\partial x^2_j}(0) \neq 0$
Since $h_{11}(0) \neq 0$ there exists a neighborhood of $0$, $N_0$ such that $h_{11}(x) \neq 0$ on $N_0$
It is then claimed that if we let $$y_1 = \sqrt{|h_{11}(x_1, ... , x_n)|} \bigg[x_1 + \sum_{i=2}^n\frac{x_ih_{i1}}{h_{11}}\bigg]$$ then $(y_1, x_2, ... x_n)$ are also local coordinates for some open set $\hat{N_0} \subset N_0$
Woah ... how did that last line come about?
The inverse function theorem applies here:
Write $y = (y_1, x_2, \cdots, x_n)^t$, and $x = (x_1,\cdots,x_n)^t$ ($t$ for transpose). Then, $$y'(0)=\left.d y\over dx\right\vert_{x=0},$$ the Jacobian matrix at $x=0$, is an upper triangular matrix, with diagonal $(\sqrt{|h_{11}(0)|},1, \cdots, 1 )$, and is therefore invertible.