Covariance matrix

Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Covariance ▷ Covariance matrix

Definition: Let $X = [X_1, \ldots, X_n]^\mathrm{T}$ be a random vector. Then, the covariance matrix of $X$ is defined as the $n \times n$ matrix in which the entry $(i,j)$ is the covariance of $X_i$ and $X_j$:

\[\label{eq:covmat} \Sigma_{XX} = \begin{bmatrix} \mathrm{Cov}(X_1,X_1) & \ldots & \mathrm{Cov}(X_1,X_n) \\ \vdots & \ddots & \vdots \\ \mathrm{Cov}(X_n,X_1) & \ldots & \mathrm{Cov}(X_n,X_n) \end{bmatrix} = \begin{bmatrix} \mathrm{E}\left[ (X_1-\mathrm{E}[X_1]) (X_1-\mathrm{E}[X_1]) \right] & \ldots & \mathrm{E}\left[ (X_1-\mathrm{E}[X_1]) (X_n-\mathrm{E}[X_n]) \right] \\ \vdots & \ddots & \vdots \\ \mathrm{E}\left[ (X_n-\mathrm{E}[X_n]) (X_1-\mathrm{E}[X_1]) \right] & \ldots & \mathrm{E}\left[ (X_n-\mathrm{E}[X_n]) (X_n-\mathrm{E}[X_n]) \right] \end{bmatrix} \; .\]

Sources:

Wikipedia (2020): "Covariance matrix"; in: Wikipedia, the free encyclopedia, retrieved on 2020-06-06; URL: https://en.wikipedia.org/wiki/Covariance_matrix#Definition.

Metadata: ID: D72 | shortcut: covmat | author: JoramSoch | date: 2020-06-06, 04:24.