In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a definite quadratic form is a real-valued quadratic form

Quadratic form

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

 over some vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

  that has the same sign (always positive or always negative) for every nonzero vector of . The definite quadratic forms correspond in one-to-one way to the (symmetric) definite bilinear form

Definite bilinear form

In mathematics, a definite bilinear form is a bilinear form B over some vector space V such that the associated quadratic formQ=B \,...

s over the same space.

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. The semidefinite quadratic forms correspond to the symmetric semidefinite bilinear forms.