Editing Covariance matrix (section)

==Definition==
Throughout this article, boldfaced unsubscripted <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> are used to refer to random vectors, and Roman subscripted <math>X_i</math> and <math>Y_i</math> are used to refer to scalar random variables.

If the entries in the [[column vector]]
<math display="block">\mathbf{X} = (X_1, X_2, \dots, X_n)^\mathsf{T}</math>
are [[random variable]]s, each with finite [[variance]] and [[expected value]], then the covariance matrix <math>\operatorname{K}_{\mathbf{X}\mathbf{X}}</math> is the matrix whose <math>(i,j)</math> entry is the [[covariance]]<ref name=KunIlPark>{{cite book | last=Park  |first = Kun Il| title=Fundamentals of Probability and Stochastic Processes with Applications to Communications| publisher=Springer | year=2018 | isbn=978-3-319-68074-3}}</ref>{{rp|p=177}}
<math display="block">\operatorname{K}_{X_i X_j} = \operatorname{cov}[X_i, X_j] = \operatorname{E}[(X_i - \operatorname{E}[X_i])(X_j - \operatorname{E}[X_j])]</math>
where the operator <math>\operatorname{E}</math> denotes the expected value (mean) of its argument.

===Conflicting nomenclatures and notations===
Nomenclatures differ. Some statisticians, following the probabilist [[William Feller]] in his two-volume book ''An Introduction to Probability Theory and Its Applications'',<ref name="Feller1971">{{cite book|author=William Feller|title=An introduction to probability theory and its applications|url=https://books.google.com/books?id=K7kdAQAAMAAJ|access-date=10 August 2012|year=1971|publisher=Wiley|isbn=978-0-471-25709-7}}</ref>  call the matrix <math>\operatorname{K}_{\mathbf{X}\mathbf{X}}</math> the '''variance''' of the random vector <math>\mathbf{X}</math>, because it is the natural generalization to higher dimensions of the 1-dimensional variance. Others call it the '''covariance matrix''', because it is the matrix of covariances between the scalar components of the vector <math>\mathbf{X}</math>.
<math display="block">
\operatorname{var}(\mathbf{X})
=
\operatorname{cov}(\mathbf{X},\mathbf{X})
=
\operatorname{E}
\left[
 (\mathbf{X} - \operatorname{E} [\mathbf{X}])
 (\mathbf{X} - \operatorname{E} [\mathbf{X}])^\mathsf{T}
\right].
</math>

Both forms are quite standard, and there is no ambiguity between them. The matrix <math>\operatorname{K}_{\mathbf{X}\mathbf{X}}</math> is also often called the '''variance-covariance matrix''', since the diagonal terms are in fact variances.

By comparison, the notation for the [[cross-covariance matrix]] ''between'' two vectors is 
<math display="block">
\operatorname{cov}(\mathbf{X},\mathbf{Y})