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.