Editing Covariance matrix (section)

==Complex random vectors==
{{further|Complex random vector#Covariance matrix and pseudo-covariance matrix}}

The [[Variance#Generalizations|variance]] of a [[complex number|complex]] ''scalar-valued'' random variable with expected value <math>\mu</math> is conventionally defined using [[complex conjugation]]:
<math display="block">
\operatorname{var}(Z)
= \operatorname{E}\left[ (Z - \mu_Z)\overline{(Z - \mu_Z)} \right],
</math>
where the complex conjugate of a complex number <math>z</math> is denoted <math>\overline{z}</math>; thus the variance of a complex random variable is a real number.

If <math>\mathbf{Z} = (Z_1,\ldots,Z_n) ^\mathsf{T}</math> is a column vector of complex-valued random variables, then the [[conjugate transpose]] <math>\mathbf{Z}^\mathsf{H}</math> is formed by ''both'' transposing and conjugating. In the following expression, the product of a vector with its conjugate transpose results in a square matrix called the '''covariance matrix''', as its expectation:<ref name=Lapidoth>{{cite book |first=Amos |last=Lapidoth |year=2009 |title=A Foundation in Digital Communication |publisher=Cambridge University Press |isbn=978-0-521-19395-5}}</ref>{{rp|p=293}}
<math display="block">
\operatorname{K}_{\mathbf{Z}\mathbf{Z}} = \operatorname{cov}[\mathbf{Z},\mathbf{Z}] =
\operatorname{E}
\left[
 (\mathbf{Z} - \boldsymbol{\mu}_\mathbf{Z})(\mathbf{Z} - \boldsymbol{\mu}_\mathbf{Z})^\mathsf{H}
\right],
</math>
The matrix so obtained will be [[Hermitian matrix|Hermitian]] [[Positive-semidefinite matrix|positive-semidefinite]],<ref>{{Cite web |url=http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/expect.html |first=Mike |last=Brookes |title=The Matrix Reference Manual}}</ref> with real numbers in the main diagonal and complex numbers off-diagonal.

;Properties
* The covariance matrix is a [[Hermitian matrix]], i.e. <math>\operatorname{K}_{\mathbf{Z}\mathbf{Z}}^\mathsf{H} = \operatorname{K}_{\mathbf{Z}\mathbf{Z}}</math>.<ref name=KunIlPark/>{{rp||page=179}}
* The diagonal elements of the covariance matrix are real.<ref name=KunIlPark/>{{rp||page=179}}

===Pseudo-covariance matrix{{anchor|Pseudo}}===
For complex random vectors, another kind of second central moment, the '''pseudo-covariance matrix''' (also called '''relation matrix''') is defined as follows:
<math display="block">
\operatorname{J}_{\mathbf{Z}\mathbf{Z}} = \operatorname{cov}[\mathbf{Z},\overline{\mathbf{Z}}] =
\operatorname{E}
\left[
 (\mathbf{Z} - \boldsymbol{\mu}_\mathbf{Z})(\mathbf{Z} - \boldsymbol{\mu}_\mathbf{Z})^\mathsf{T}
\right]
</math>

In contrast to the covariance matrix defined above, Hermitian transposition gets replaced by transposition in the definition.
Its diagonal elements may be complex valued; it is a [[complex symmetric matrix]].