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In mathematics, a definite bilinear form is a bilinear form B such that## See also

- 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=R^{2}, and consider the bilinear form

- $B(x,\; y)=c\_1x\_1y\_1+c\_2x\_2y\_2$

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.

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Last updated on Friday June 27, 2008 at 06:40:17 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday June 27, 2008 at 06:40:17 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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