Definite bilinear form

Definite bilinear form

In mathematics, a definite bilinear form is a bilinear form B over some vector space V (with real or complex scalar field) such that the associated quadratic form

$Q(x)=B(x, x) \,$

is definite, that is, has a real value with the same sign (positive or negative) for all non-zero x. According to that sign, B is called positive definite or negative definite. If Q takes both positive and negative values, the bilinear form B is called indefinite.

If B(x, x) ≥ 0 for all x, B is said to be positive semidefinite. Negative semidefinite bilinear forms are defined similarly.

Example

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.

Properties

When the scalar field of V is the complex numbers, the function Q defined by $Q (x) = B (x, x)$ is real-valued only if B is Hermitian, that is, if B(x, y) is always the complex conjugate of B(y, x).

A self-adjoint operator A on an inner product space is positive definite if

(x, Ax) > 0 for every nonzero vector x.

See also

Positive definite function
Positive definite matrix

Categories:

Multilinear algebra

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