covariance matrix

Let $𝐗={(X_1,\mathrm{\dots },X_n)}^T$ be a random vector. Then the covariance matrix of $𝐗$, denoted by $\mathrm{𝐂𝐨𝐯}(𝐗)$, is $\{Cov(X_i,X_j)\}$. The diagonals of $\mathrm{𝐂𝐨𝐯}(𝐗)$ are $Cov(X_i,X_i)=Var[X_i]$. In matrix notation,

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

The covariance matrix is symmetric and if the $X_i$’s are independent, identically distributed (iid) with variance ${𝝈}^2$, then

$$\mathrm{𝐂𝐨𝐯}(𝐗)={𝝈}^2𝐈.$$