# Definite bilinear form

(Redirected from Positive-definite)

Jump to navigation
Jump to search
In mathematics, a **definite bilinear form** *B* is one for which

*B*(*v*,*v*)

has a fixed sign (positive or negative) when it is not 0.

To give a formal definition, let *K* be one of the fields **R** (real numbers) or **C** (complex numbers). Suppose that *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*. If it is greater than or equal to zero, we say *B* is **positive semidefinite**. Similarly for **negative definite** and **negative semidefinite**. If it is otherwise unconstrained, we say *B* is **indefinite**.

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.