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Diagonal Matrix Independent Symmetric Matrix Variance Random Vector Independent Identically Distributed

1	Diagonal
1	Matrix
2	Independent
3	Variance
3	Symmetric Matrix
5	Independent Identically Distributed
6	Random Vector

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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 $\lbrace Cov(X_i,X_j) \rbrace$ . The diagonals of $\mathbf{Cov(X)}$ are . In matrix notation,

$\displaystyle \mathbf{Cov(X)}=\begin{pmatrix}Var[X_1] & \cdots & Cov(X_1,X_n) \\ \vdots & & \vdots \\ Cov(X_n,X_1) & \cdots & Var[X_n] \end{pmatrix}.$

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

$\displaystyle \begin{pmatrix}E[{X_1}^2]-E[X_1]^2 & \cdots & E[X_1X_n]-E[X_1]E[X... ...[X_n]^2 \end{pmatrix} = \mathbf{E\Big[\big(X-E[X]\big)\big(X-E[X]\big)^T\Big]}.$

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

$\displaystyle \mathbf{Cov(X)}=\boldsymbol{\sigma}^2\mathbf{I}.$