# Positive definite

Let

*K* be the

field **R** or

**C**,

*V* is a

vector space over

*K*, and

*B* :

*V* ×

*V* →

*K* is a

bilinear map which is Hermitian in the sense that

*B*(

*x*,

*y*) is always the complex conjugate of

*B*(

*y*,

*x*). Then

*B* is

**positive-definite** if

*B*(

*x*,

*x*) > 0 for every nonzero

*x* in

*V*.

A self-adjoint operator *A* on an inner product space is **positive-definite** if (*x*, *Ax*) > 0 for every nonzero vector *x*.

See in particular positive-definite matrix.