Definite bilinear form

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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 × VK

is a bilinear map which is Hermitian in the sense that B(xy) is always the complex conjugate of B(yx). 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.

See also