Editing Definite matrix (section)

== Ramifications ==
It follows from the above definitions that a matrix is positive-definite [[if and only if]] it is the matrix of a [[positive-definite quadratic form]] or [[Hermitian form]]. In other words, a matrix is positive-definite if and only if it defines an [[inner product]].

Positive-definite and positive-semidefinite matrices can be characterized in many ways, which may explain the importance of the concept in various parts of mathematics. A matrix {{mvar|M}} is positive-definite if and only if it satisfies any of the following equivalent conditions.
* <math>M</math> is [[congruent matrices|congruent]] with a [[diagonal matrix]] with positive  real entries. 
* <math>M</math> is symmetric or Hermitian, and all its [[eigenvalue]]s are real and positive.
* <math>M</math> is symmetric or Hermitian, and all its leading [[principal minor]]s are positive.
* There exists an [[invertible matrix]] <math>B</math> with conjugate transpose <math>B^*</math> such that <math>M = B^* B.</math>
A matrix is positive semi-definite if it satisfies similar equivalent conditions where "positive" is replaced by "nonnegative", "invertible matrix" is replaced by "matrix", and the word "leading" is removed.

Positive-definite and positive-semidefinite real matrices are at the basis of [[convex optimization]], since, given a [[function of several real variables]] that is twice [[differentiable function|differentiable]], then if its [[Hessian matrix]] (matrix of its second partial derivatives) is positive-definite at a point <math>p,</math> then the function is [[convex function|convex]] near {{mvar|p}}, and, conversely, if the function is convex near <math>p,</math> then the Hessian matrix is positive-semidefinite at <math>p.</math>

The set of positive definite matrices is an [[Open set|open]] [[convex cone]], while the set of positive semi-definite matrices is a [[closed set|closed]] convex cone.<ref>
{{cite book
 |last1=Boyd         |first1=Stephen
 |last2=Vandenberghe |first2=Lieven
 |date=2004-03-08
 |title=Convex Optimization
 |publisher=Cambridge University Press
 |isbn=978-0-521-83378-3
 |doi=10.1017/cbo9780511804441
}}
</ref>