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
Canonical name	CovarianceMatrix
Date of creation	2013-03-22 14:27:23
Last modified on	2013-03-22 14:27:23
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	8
Author	CWoo (3771)
Entry type	Definition
Classification	msc 62H99
Synonym	variance covariance matrix