Statement: If \(p\in {\mathbf{R}}^n\) is a critical point of \(f: {\mathbf{R}}^n \to {\mathbf{R}}\) 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 is well-defined.