Definite matrix: Difference between revisions

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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}^\top M \mathbf{x}\ </math> and <math>\ \mathbf{z}^* M \mathbf{z}\ </math> are required to be positive ''or zero'' (that is, not negativenonnegative). '''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''.

== Ramifications ==