Editing Definite matrix (section)

=== Local extrema ===
A general [[quadratic form]] <math>f(\mathbf{x})</math> on <math>n</math> real variables <math>x_1, \ldots, x_n</math> can always be written as <math>\mathbf{x}^\mathsf{T} M \mathbf{x}</math> where <math>\mathbf{x}</math> is the column vector with those variables, and <math>M</math> is a symmetric real matrix. Therefore, the matrix being positive definite means that <math>f</math> has a unique minimum (zero) when <math>\mathbf{x}</math> is zero, and is strictly positive for any other <math>\mathbf{x}.</math>

More generally, a twice-differentiable real function <math>f</math> on <math>n</math> real variables has local minimum at arguments <math>x_1, \ldots, x_n</math> if its [[gradient]] is zero and its [[Hessian matrix|Hessian]] (the matrix of all second derivatives) is positive semi-definite at that point. Similar statements can be made for negative definite and semi-definite matrices.