Editing Definite matrix (section)

{{Short description|Property of a mathematical matrix}}
{{hatnote|Not to be confused with [[Positive matrix]] and [[Totally positive matrix]].}}
{{use dmy dates|date=June 2024}}

In [[mathematics]], a symmetric matrix <math>M</math> with [[real number|real]] entries is  '''positive-definite''' if the real number <math>\mathbf{x}^\mathsf{T} M \mathbf{x}</math> is positive for every nonzero real [[column vector]] <math>\mathbf{x},</math> where <math>\mathbf{x}^\mathsf{T}</math> is the [[row vector]] [[transpose]] of <math>\mathbf{x}.</math><ref>
{{cite book
 |first = Adriaan  |last = van den Bos
 |date = March 2007
 |section = Appendix C: Positive semidefinite and positive definite matrices
 |title = Parameter Estimation for Scientists and Engineers |edition=online
 |publisher = John Wiley & Sons
 |isbn = 978-047-017386-2
 |pages = 259–263
 |doi = 10.1002/9780470173862 |doi-access=
 |section-url = https://www.doi.org/10.1002/9780470173862.app3 |type = .pdf
}} Print ed. {{ISBN|9780470147818}}
</ref>
More generally, a [[Hermitian matrix]] (that is, a [[complex matrix]] equal to its [[conjugate transpose]])  is '''positive-definite''' if the real number <math>\mathbf{z}^* M \mathbf{z}</math> is positive for every nonzero complex column vector <math>\mathbf{z},</math> where <math>\mathbf{z}^*</math> denotes the conjugate transpose of <math>\mathbf{z}.</math>

'''Positive semi-definite''' matrices are defined similarly, except that the scalars <math>\mathbf{x}^\mathsf{T} M \mathbf{x}</math> and <math>\mathbf{z}^* M \mathbf{z}</math> are required to be positive ''or zero'' (that is, nonnegative). '''Negative-definite''' and '''negative semi-definite''' matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called ''indefinite''.

Some authors use more general definitions of definiteness, permitting the matrices to be non-symmetric or non-Hermitian. The properties of these generalized definite matrices are explored in {{alink|Extension for non-Hermitian square matrices}}, below, but are not the main focus of this article.