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 × V → K

is a bilinear form which is Hermitian in the sense that B(x, y) is always the complex conjugate of B(y, x). 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

Failed to parse (Missing texvc executable; please see math/README to configure.): B(x, y)=c_1x_1y_1+c_2x_2y_2

where Failed to parse (Missing texvc executable; please see math/README to configure.): x=(x_1, x_2) , Failed to parse (Missing texvc executable; please see math/README to configure.): y=(y_1, y_2) , and Failed to parse (Missing texvc executable; please see math/README to configure.): c_1

and Failed to parse (Missing texvc executable; please see math/README to configure.): c_2
are constants. If Failed to parse (Missing texvc executable; please see math/README to configure.): c_1>0
and Failed to parse (Missing texvc executable; please see math/README to configure.): c_2>0

, the bilinear form Failed to parse (Missing texvc executable; please see math/README to configure.): B

is positive definite. If one of the constants is positive and the other is zero, then Failed to parse (Missing texvc executable; please see math/README to configure.): B
is positive semidefinite. If Failed to parse (Missing texvc executable; please see math/README to configure.): c_1>0
and Failed to parse (Missing texvc executable; please see math/README to configure.): c_2<0

, then Failed to parse (Missing texvc executable; please see math/README to configure.): B

is indefinite.

Given a Hermitian bilinear form Failed to parse (Missing texvc executable; please see math/README to configure.): B , the function

Failed to parse (Missing texvc executable; please see math/README to configure.): Q(x)=B(x, x)

is a quadratic form. The definitions of definiteness for Failed to parse (Missing texvc executable; please see math/README to configure.): B

are then transferred to corresponding definitions for Failed to parse (Missing texvc executable; please see math/README to configure.): 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

• Positive definite function
• Positive definite matrixde:Definitheit

es:Positivo definido