Definite quadratic form
In
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.[1]
Associated symmetric bilinear form
Quadratic forms correspond one-to-one to symmetric bilinear forms over the same space.[2] 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
Examples
As an example, let , and consider the quadratic form
where and c1 and c2 are constants. If c1 > 0 and c2 > 0 , 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 c1 > 0 and c2 < 0 , or vice versa, then Q is indefinite and sometimes evaluates to a positive number and sometimes to a negative number. If c1 < 0 and c2 < 0 , 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 x1·x2:
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. If the general quadratic form above is equated to 0, the resulting equation is that of an ellipse if the quadratic form is positive or negative-definite, a hyperbola if it is indefinite, and a parabola if
The square of the
In two dimensions this means that the distance between two points is the square root of the sum of the squared distances along the axis and the axis.
Matrix form
A quadratic form can be written in terms of matrices as
where x is any n×1 Cartesian vector in which at least one element is not 0; A is an n × n
Positive or negative-definiteness or semi-definiteness, or indefiniteness, of this quadratic form is equivalent to
Optimization
Definite quadratic forms lend themselves readily to
where b is an n×1 vector of constants. The
giving
assuming A is
An important example of such an optimization arises in
See also
- Isotropic quadratic form
- Positive-definite function
- Positive-definite matrix
- Polarization identity
Notes
- ^ Milnor & Husemoller 1973, p. 61.
- ^ This is true only over a field of characteristic other than 2, but here we consider only ordered fields, which necessarily have characteristic 0.
References
- Kitaoka, Yoshiyuki (1993). Arithmetic of quadratic forms. Cambridge Tracts in Mathematics. Vol. 106. Cambridge University Press. Zbl 0785.11021.
- ISBN 978-0-387-95385-4.
- Zbl 0292.10016.