# Quadratic form

### From Glossary

The function, where is an n-vector and is an matrix. Its gradient is Typically, is presumed symmetric, in which case its gradient is and its eigenvalues are real.

- The quadratic form is:
*positive semi-definite*if for all in which case its eigenvalues are non-negative, and the quadratic form is a convex function.*positive definite*if for all nonzero in which case its eigenvalues are positive, and the quadratic form is a strongly convex function.*negative semi-definite*if for all in which case its eigenvalues are non-positive, and the quadratic form is a concave function.*negative definite*if for all nonzero in which case its eigenvalues are negative, and the quadratic form is a strongly concave function.*indefinite*if it is none of the above.