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.
Read more about this topic: Positive-definite Matrix
Famous quotes containing the word forms:
“The catalogue of forms is endless: until every shape has found its city, new cities will continue to be born. When the forms exhaust their variety and come apart, the end of cities begins.”
—Italo Calvino (1923–1985)