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, vtAv > 0.
Like positive semidefiniteness does for positive real numbers, negative semidefiniteness extends some properties of "negativeness" of real numbers. Given a function f:Rn->R, if ∇2f(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.