Definite matrix: Difference between revisions

Revision as of 03:08, 1 February 2024

Revision as of 03:08, 1 February 2024

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{{hatnote|Not to be confused with [[Positive matrix]] and [[Totally positive matrix]].}}

In [[mathematics]], a symmetric matrix <math>M</math> with [[real number|real]] entries is '''positive-definite''' if the real numb!ernumber <math>z^{\operatorname{T}}Mz</math> is positive for every nonzero real [[column vector]] <math>z,</math> where <math>z^\operatorname{T}</math> is the [[transpose]] of {{nowrap|<math>z</math>.}}<ref>{{cite journal|doi=10.1002/9780470173862.app3 | title=Appendix C: Positive Semidefinite and Positive Definite Matrices | journal=Parameter Estimation for Scientists and Engineers | pages=259–263| doi-access= }}</ref> More generally, a [[Hermitian matrix]] (that is, a [[complex matrix]] equal to its [[conjugate transpose]]) is

'''positive-definite''' if the real number <math>z^* Mz</math> is positive for every nonzero complex column vector <math>z,</math> where <math>z^*</math> denotes the conjugate transpose of <math>z.</math>