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 = (x1,x2), y = (y1,y2), and c1 and c2 are constants. If c1 > 0 and c2 > 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 c1 > 0 and c2 < 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(xy) is always the complex conjugate of B(yx).

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

Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Definite quadratic form — In mathematics, a definite quadratic form is a real valued quadratic form over some vector space V that has the same sign (always positive or always negative) for every nonzero vector of V. The definite quadratic forms correspond in one to one… …   Wikipedia

  • form — I. noun Etymology: Middle English forme, from Anglo French furme, forme, from Latin forma form, beauty Date: 13th century 1. a. the shape and structure of something as distinguished from its material b. a body (as of a person) especially in its… …   New Collegiate Dictionary

  • Positive definite — In mathematics, positive definite may refer to: * positive definite matrix * positive definite function ** positive definite function on a group * positive definite bilinear form …   Wikipedia

  • Quadratic form — In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example, is a quadratic form in the variables x and y. Quadratic forms occupy a central place in various branches of mathematics, including… …   Wikipedia

  • Positive-definite matrix — In linear algebra, a positive definite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positive definite symmetric bilinear form (or a sesquilinear form in the complex case). The… …   Wikipedia

  • Killing form — In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. In an example of Stigler s law of eponymy, the Killing form was actually invented… …   Wikipedia

  • Isotropic quadratic form — In mathematics, a quadratic form over a field F is said to be isotropic if there is a non zero vector on which it evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if q is a quadratic form on a vector space V over F …   Wikipedia

  • Intersection form (4-manifold) — In mathematics, the intersection form of an oriented compact 4 manifold is a special symmetric bilinear form on the 2nd cohomology group of the 4 manifold. It reflects much of the topology of the 4 manifolds, including information on the… …   Wikipedia

  • Real form (Lie theory) — Lie groups …   Wikipedia

  • Riemann form — In mathematics, a Riemann form in the theory of abelian varieties and modular forms, is the following data:* A lattice Λ in a complex vector space Cg. * An alternating bilinear form α from Λ to the integers satisfying the following two conditions …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”