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

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