Probability and Statistics

Revise

Covariance Matrix

Square matrix that represents the covariance between each pair of elements in a given multivariate random variable.

Covariance matrix for a two-dimensional random variable:

$$\Sigma =\left[\begin{array}{cc}{\sigma }_{xx}& {\sigma }_{xy}\\ {\sigma }_{yx}& {\sigma }_{yy}\end{array}\right]=\left[\begin{array}{cc}{\sigma }_x^2& {\sigma }_{xy}\\ {\sigma }_{yx}& {\sigma }_y^2\end{array}\right]=\left[\begin{array}{cc}VAR(\mathbf{x})& COV(\mathbf{x},\mathbf{y})\\ COV(\mathbf{y},\mathbf{x})& VAR(\mathbf{y})\end{array}\right]$$

Note: Remember that the off-diagonal entries of the covariance matrix are equal since $COV(\mathbf{x},\mathbf{y})=COV(\mathbf{y},\mathbf{x})$. If $x$ and $y$ are uncorrelated, they are equal to $0$ so the off-diagonal entries will be $0$ as well.

Related

• Multivariate Kalman Filter
• Eigenvalues and Eigenvectors