Editing Symmetric matrix (section)

== Hessian ==
Symmetric <math>n \times n</math> matrices of real functions appear as the [[Hessian matrix|Hessians]] of twice differentiable functions of <math>n</math> real variables (the continuity of the second derivative is not needed, despite common belief to the opposite<ref>{{Cite book |last=Dieudonné |first=Jean A. |title=Foundations of Modern Analysis |publisher=Academic Press |year=1969 |chapter=Theorem (8.12.2) |page=180 |isbn=0-12-215550-5 |oclc=576465}}</ref>). 

Every [[quadratic form]] <math>q</math> on <math>\mathbb{R}^n</math> can be uniquely written in the form <math>q(\mathbf{x}) = \mathbf{x}^\textsf{T} A \mathbf{x}</math> with a symmetric <math>n \times n</math> matrix <math>A</math>. Because of the above spectral theorem, one can then say that every quadratic form, up to the choice of an orthonormal basis of <math>\R^n</math>, "looks like"
<math display="block">q\left(x_1, \ldots, x_n\right) = \sum_{i=1}^n \lambda_i x_i^2</math>
with real numbers <math>\lambda_i</math>. This considerably simplifies the study of quadratic forms, as well as the study of the level sets <math>\left\{ \mathbf{x} : q(\mathbf{x}) = 1 \right\}</math> which are generalizations of [[conic section]]s.

This is important partly because the second-order behavior of every smooth multi-variable function is described by the quadratic form belonging to the function's Hessian; this is a consequence of [[Taylor's theorem]].