Editing Definite matrix (section)

=== Notation ===
If a Hermitian matrix <math>M</math> is positive semi-definite, one sometimes writes <math>M \succeq 0</math> and if <math>M</math> is positive-definite one writes <math>M \succ 0.</math> To denote that <math>M</math> is negative semi-definite one writes <math>M \preceq 0</math> and to denote that <math>M</math> is negative-definite one writes <math>M \prec 0.</math>

The notion comes from [[functional analysis]] where positive semidefinite matrices define [[positive operator]]s. If two matrices <math>A</math> and <math>B</math> satisfy <math>B - A \succeq 0,</math> we can define a [[Partially ordered set#Non-strict partial order|non-strict partial order]] <math>B \succeq A</math> that is [[Reflexive relation|reflexive]], [[Antisymmetric relation|antisymmetric]], and [[Transitive relation|transitive]]; It is not a [[total order]], however, as <math>B - A,</math> in general, may be indefinite.

A common alternative notation is <math>M \geq 0,</math> <math>M > 0,</math> <math>M \leq 0,</math> and <math>M < 0</math> for positive semi-definite and positive-definite, negative semi-definite and negative-definite matrices, respectively. This may be confusing, as sometimes [[nonnegative matrix|nonnegative matrices]] (respectively, nonpositive matrices) are also denoted in this way.