Statement: 
If $p\in \RR^n$ is a critical point of $f: \RR^n \to \RR$ such that the Hessian $H_f(p)$ is a non-degenerate [standard form](bilinear form](standard%20form](bilinear%20form), then $f$ is locally a Morse function (standard form).

 Moreover, after diagonalizing $H_f$, the index is given by the difference in the numbers of positive/negatives on the diagonal.
 
 Nondegenerate critical points have standard forms $\sum \pm x_i^2$, so the [index of a Morse function](index%20of%20a%20Morse%20function) is well-defined.