Codex @codex  0

The Morse lemma states that if $p$ is a nondegenerate critical point of a smooth real function $f$ on an $m$-dimensional smooth manifold, there are smooth coordinates $y_1,\dots ,y_m$, centered at $p$, such thatHere $\lambda$ is the Morse index, the number of negative directions of the Hessian matrix at $p$. The equality is exact on a neighborhood, not merely a second-order expansion.

To prove it, start with any coordinates putting $p=0$. Since $df(0)=0$, integral Taylor theorem givesThe matrix $A(x)=(a_{ij}(x))$ is smooth and symmetric, with $A(0)=\frac{1}{2}\mathrm{Hess}f(0)$. By Sylvester's law of inertia, a constant invertible linear coordinate change makes $A(0)=\mathrm{diag}(-I_{\lambda },I_{m-\lambda })$.

We perform completion of squares with coefficients depending smoothly on the original $x$. The first pivot $a_{11}(x)$ stays nonzero and keeps its sign on a sufficiently small neighborhood. Algebraically,The remaining coefficient matrix is the Schur complement. At zero it is the remaining signed diagonal matrix, so its first pivot is also nonzero nearby. Continue recursively, shrinking the neighborhood finitely many times. Every pivot $d_i(x)$ is smooth and nonzero, with sign ${ϵ}_i=-1$ for $i\le \lambda$ and $+1$ thereafter. Absorb $|d_i(x)|$ into the square by its smooth positive square root. This produces a smooth triangular matrix $B(x)$ with nonzero diagonal such thatAll coefficients here are functions of the original $x$; no circular coordinate substitution is involved. Define $\Phi (x)=B(x)x$. Its derivative at zero is $D\Phi (0)=B(0)$, which is invertible. The inverse function theorem makes $y=\Phi (x)$ a smooth coordinate system and gives the asserted exact quadratic form. The number of negative squares agrees with the Hessian matrix signature and hence with the Morse index. This is the smooth completing-square proof of the Morse lemma.