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| − | A [[Quadratic form|quadratic form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084060/s0840601.png" /> over an ordered field which represents 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 (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084060/s0840602.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084060/s0840603.png" />), in the latter case — a non-positive definite quadratic form (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084060/s0840604.png" />). Most frequently one considers semi-definite forms over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084060/s0840605.png" /> of real numbers. For the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084060/s0840606.png" /> a similar definition yields the concept of (non-negative and non-positive) semi-definite Hermitian quadratic forms (see [[Hermitian form|Hermitian form]]).
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| − | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084060/s0840607.png" /> is a symmetric bilinear (cf. [[Bilinear form|Bilinear form]]) or Hermitian form such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084060/s0840608.png" /> is a semi-definite form, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084060/s0840609.png" /> is sometimes also called a (non-negative or non-positive) semi-definite form. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084060/s08406010.png" /> is a quadratic or Hermitian semi-definite form in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084060/s08406011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084060/s08406012.png" /> is a subspace, identical with the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084060/s08406013.png" />, and the given form naturally induces a positive-definite or negative-definite form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084060/s08406014.png" />.
| + | 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 number]]s. For the field $\mathbb C$ of [[complex number]]s 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'''. |
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| − | | + | 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 $V/N$. |
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| − | ====Comments==== | |
| − | Instead of "non-negative definite" one also says positive semi-definite, and instead of "non-positive definite" also negative semi-definite.
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Latest revision as of 17:54, 18 January 2016
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 $V/N$.
How to Cite This Entry:
Semi-definite form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-definite_form&oldid=11755
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article