# covariance matrix

Let $\mathbf{X}=(X_{1},\ldots,X_{n})^{T}$ be a random vector. Then the covariance matrix of $\mathbf{X}$, denoted by $\mathbf{Cov(X)}$, is $\{Cov(X_{i},X_{j})\}$. The diagonals of $\mathbf{Cov(X)}$ are $Cov(X_{i},X_{i})=Var[X_{i}]$. In matrix notation,

 $\mathbf{Cov(X)}=\begin{pmatrix}Var[X_{1}]&\cdots&Cov(X_{1},X_{n})\\ \vdots&&\vdots\\ Cov(X_{n},X_{1})&\cdots&Var[X_{n}]\end{pmatrix}.$

It is easily seen that $\mathbf{Cov(X)}=\mathbf{Var[X]}$ via

 $\begin{pmatrix}E[{X_{1}}^{2}]-E[X_{1}]^{2}&\cdots&E[X_{1}X_{n}]-E[X_{1}]E[X_{n% }]\\ \vdots&&\vdots\\ E[X_{n}X_{1}]-E[X_{n}]E[X_{1}]&\cdots&E[{X_{n}}^{2}]-E[X_{n}]^{2}\end{pmatrix}% =\mathbf{E\Big{[}\big{(}X-E[X]\big{)}\big{(}X-E[X]\big{)}^{T}\Big{]}}.$

The covariance matrix is symmetric and if the $X_{i}$’s are independent, identically distributed (iid) with variance $\boldsymbol{\sigma}^{2}$, then

 $\mathbf{Cov(X)}=\boldsymbol{\sigma}^{2}\mathbf{I}.$
Title covariance matrix CovarianceMatrix 2013-03-22 14:27:23 2013-03-22 14:27:23 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 62H99 variance covariance matrix