Definition (Covariance matrix)

Given a random vector $X^{n}=(X_{1},\cdots,X_{n})$, the covariance matrix of $X^{n}$, $K_{X_{i},X_{j}}$, is defined as follows $$K_{X_{i},X_{j}}=\begin{pmatrix} \mathrm{Var}(X_{1}) & \mathrm{Cov}(X_{1},X_{2}) & \cdots & \mathrm{Cov}(X_{1},X_{n}) \\ \mathrm{Cov}(X_{2},X_{1}) & \mathrm{Var}(X_{2}) & \cdots & \mathrm{Cov}(X_{2},X_{n}) \\ \vdots & \vdots & \ddots & \vdots \\ \mathrm{Cov}(X_{n},X_{1}) & \mathrm{Cov}(X_{n},X_{2}) & \cdots & \mathrm{Var}(X_{n})\end{pmatrix}$$