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Concept
Definite quadratic form
Summary
In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every non-zero vector of V. According to that sign, the quadratic form is called positive-definite or negative-definite.
A semidefinite (or semi-definite) quadratic form is defined in much the same way, except that "always positive" and "always negative" are replaced by "never negative" and "never positive", respectively. In other words, it may take on zero values for some non-zero vectors of V.
An indefinite quadratic form takes on both positive and negative values and is called an isotropic quadratic form.
More generally, these definitions apply to any vector space over an ordered field.
Quadratic forms correspond one-to-one to symmetric bilinear forms over the same space. A symmetric bilinear form is also described as definite, semidefinite, etc. according to its associated quadratic form. A quadratic form Q and its associated symmetric bilinear form B are related by the following equations:
The latter formula arises from expanding
As an example, let , and consider the quadratic form
where and c_1 and c_2 are constants. If and the quadratic form Q is positive-definite, so Q evaluates to a positive number whenever If one of the constants is positive and the other is 0, then Q is positive semidefinite and always evaluates to either 0 or a positive number. If and or vice versa, then Q is indefinite and sometimes evaluates to a positive number and sometimes to a negative number. If and the quadratic form is negative-definite and always evaluates to a negative number whenever And if one of the constants is negative and the other is 0, then Q is negative semidefinite and always evaluates to either 0 or a negative number.
In general a quadratic form in two variables will also involve a cross-product term in x_1·x_2:
This quadratic form is positive-definite if and negative-definite if and and indefinite if It is positive or negative semidefinite if with the sign of the semidefiniteness coinciding with the sign of
This bivariate quadratic form appears in the context of conic sections centered on the origin.
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