Positive-definite Matrix - Quadratic Forms

Quadratic Forms

The (purely) quadratic form associated with a real matrix M is the function Q from to such that for all x. It turns out that the matrix M is positive definite if and only if it is symmetric and its quadratic form is a strictly convex function.

More generally, any quadratic function from to can be written as where is a symmetric n×n matrix, b is a real n-vector, and c a real constant. This quadratic function is strictly convex, and hence has a unique finite global minimum, if and only if M is positive definite. For this reason, positive definite matrices play an important role in optimization problems.