# 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.^{[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:

## Example

As an example, let , and consider the quadratic form

where *x* = (*x*_{1}, *x*_{2}) and *c*_{1} and *c*_{2} are constants. If *c*_{1} > 0 and *c*_{2} > 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 *c*_{1} > 0 and *c*_{2} < 0, then *Q* is indefinite.

## See also

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

## References

- ↑ 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.

- Kitaoka, Yoshiyuki (1993).
*Arithmetic of quadratic forms*. Cambridge Tracts in Mathematics.**106**. Cambridge University Press. ISBN 0-521-40475-4. Zbl 0785.11021. - Lang, Serge (2004),
*Algebra*, Graduate Texts in Mathematics,**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.**73**. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016.