bilinear form

Definite bilinear form

In mathematics, a definite bilinear form is a bilinear form B such that

B(x, x)

has a fixed sign (positive or negative) when x 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 form which is Hermitian in the sense that B(xy) is always the complex conjugate of B(yx). Then B is called positive definite if

B(x, x) > 0

for every nonzero x in V. If B(x, x) ≥ 0 for all x, B is said to be positive semidefinite. Negative definite and negative semidefinite bilinear forms are defined similarly. If B(x, x) takes both positive and negative values, it is called indefinite.

As an example, let V=R2, and consider the bilinear form

B(x, y)=c_1x_1y_1+c_2x_2y_2
where x=(x_1, x_2), y=(y_1, y_2), and c_1 and c_2 are constants. If c_1>0 and c_2>0, the bilinear form B is positive definite. If one of the constants is positive and the other is zero, then B is positive semidefinite. If c_1>0 and c_2<0, then B is indefinite.

Given a Hermitian bilinear form B, the function

Q(x)=B(x, x)

is a quadratic form. The definitions of definiteness for B are then transferred to corresponding definitions for Q.

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

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