Definite quadratic form

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 nonzero 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 the same way, except that "positive" and "negative" are replaced by "not negative" and "not positive", respectively. An indefinite quadratic form is one that takes on both positive and negative values.

More generally, the definition applies to a vector space over an ordered field.

Associated symmetric bilinear form

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:

$\,Q(x)=B(x,x)$

$\,B(x,y)=B(y,x)={\tfrac {1}{2}}(Q(x+y)-Q(x)-Q(y))$

Examples

As an example, let $V=\mathbb {R} ^{2}$, and consider the quadratic form

$Q(x)=c_{1}{x_{1}}^{2}+c_{2}{x_{2}}^{2}$

where x = (x₁, x₂) and c₁ and c₂ are constants. If c₁ > 0 and c₂ > 0, the quadratic form Q is positive definite. If one of the constants is positive and the other is zero, then Q is positive semidefinite. If c₁ > 0 and c₂ < 0, then Q is indefinite.

In general a quadratic form in two variables will also involve a cross-product term in x₁x₂:

$Q(x)=c_{1}{x_{1}}^{2}+c_{2}{x_{2}}^{2}+2c_{3}x_{1}x_{2}.$

This quadratic form is positive definite if $c_{1}>0$ and $c_{1}c_{2}-{c_{3}}^{2}>0,$ negative definite if $c_{1}<0$ and $c_{1}c_{2}-{c_{3}}^{2}>0,$ and indefinite if $c_{1}c_{2}-{c_{3}}^{2}<0.$ It is positive or negative semidefinite if $c_{1}c_{2}-{c_{3}}^{2}=0,$ with the sign of the semidefiniteness coinciding with the sign of $c_{1}.$

Matrix form

A quadratic form can be written in terms of matrices as

$x^{T}Ax$

where x is any n×1 Cartesian vector $(x_{1},\cdots ,x_{n})^{T}$ in which not all elements are 0, superscript ^T denotes a transpose, and A is an n×n symmetric matrix. If A is diagonal this is equivalent to a non-matrix form containing solely terms involving squared variables; but if A has any non-zero off-diagonal elements, the non-matrix form will also contain some terms involving products of two different variables.

Positive or negative definiteness or semi-definiteness, or indefiniteness, of this quadratic form is equivalent to the same property of A, which can be checked by considering all eigenvalues of A or by checking the signs of all of its principal minors.

See also

• Anisotropic quadratic form
• Positive definite function
• Positive definite matrix
• Polarization identity

References

• Kitaoka, Yoshiyuki (1993). Arithmetic of quadratic forms. Cambridge Tracts in Mathematics. Vol. 106. Cambridge University Press. ISBN 0-521-40475-4. Zbl 0785.11021.
• Lang, Serge (2004), Algebra, Graduate Texts in Mathematics, vol. 211 (Corrected fourth printing, revised third ed.), New York: Springer-Verlag, p. 578, ISBN 978-0-387-95385-4
• Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016.