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
is definite, that is, has a real value with the same sign (positive or negative) for all nonzero 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=R^{2}, and consider the bilinear form
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 realvalued only if B is Hermitian, that is, if B(x, y) is always the complex conjugate of B(y, x).
A selfadjoint 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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