Semi-definite form

2020 Mathematics Subject Classification: Primary: 15A63 [MSN][ZBL]

A quadratic form $q$ on a vector space over an ordered field for which the set of values consists of either only non-negative field elements or only non-positive field elements. In the former case the form is said to be a non-negative definite or positive semi-definite form ($q(x) \ge 0$ for all $x$), in the latter case — a non-positive definite or negative semi-definite quadratic form ($q(x) \le 0$). Most frequently one considers semi-definite forms over the field $\mathbb R$ of real numbers. For the field $\mathbb C$ of complex numbers a similar definition yields the concept of semi-definite Hermitian quadratic forms (see Hermitian form). If in addition $q(x) \ne 0$ for $x \ne 0$ the form is definite.

If $b$ is a symmetric bilinear (cf. Bilinear form) or Hermitian form such that $q(x) = b(x,x)$ is a semi-definite form, then $b$ may be called a (non-negative or non-positive) semi-definite form. If $q$ is a quadratic or Hermitian semi-definite form on a vector space $V$, then $N = \{x \in V : q(x) = 0\}$ is a subspace, identical with the kernel of $q$, and the given form naturally induces a positive-definite or negative-definite form on $V/N$.