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.

1 Example
2 Properties
3 See also
4 References

Example

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

$B(x, y)=c_1x_1y_1%2Bc_2x_2y_2 \,$

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.

References

Nathanael Leedom Ackerman (2006) Lecture notes Math 371, Positive definite bilinear form is definition 0.5.0.7, weblink from University of California, Berkeley.

Definite bilinear form

Contents

Example

Properties

See also

References