A

bilinear form f(u,v) is called

*positive semidefinite* if for all

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

Since a quadratic form is represented by some square matrix A, we can immediately "lower" the property to matrices: A is *positive semidefinite* iff for all vectors v, v^{t}Av > 0.

Positive semidefiniteness extends some properties of "positiveness" of real numbers. For instance, the second derivative of a function f:**R**^{n}->**R** is an n*n matrix. If ∇^{2}f(x) is positive semidefinite, then f is (locally) convex near x. In one variable, you already know this: if f:**R**->**R** satisfies f''(x)>0, then f is convex near x.