A

bilinear form f(u,v) is called

*negative semidefinite* if for all

vectors v, f(v,v)<0.

In vector spaces of finite dimension, a quadratic form is represented by some square matrix A, and we can immediately "lower" the property to matrices: A is *negative semidefinite* iff for all vectors v, v^{t}Av > 0.

Like positive semidefiniteness does for positive real numbers, negative semidefiniteness extends some properties of "negativeness" of real numbers. Given a function f:**R**^{n}->**R**, if ∇^{2}f(x) is negative semidefinite, then f is (locally) concave near x. In one variable, you already know this: if f:**R**->**R** satisfies f''(x)<0, then f is concave near x.