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

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

\[\label{eq:corrmat} \mathrm{C}_{XX} = \begin{bmatrix} \mathrm{Corr}(X_1,X_1) & \ldots & \mathrm{Corr}(X_1,X_n) \\ \vdots & \ddots & \vdots \\ \mathrm{Corr}(X_n,X_1) & \ldots & \mathrm{Corr}(X_n,X_n) \end{bmatrix} = \begin{bmatrix} \frac{\mathrm{E}\left[(X_1-\mathrm{E}[X_1]) (X_1-\mathrm{E}[X_1])\right]}{\sigma_{X_1} \, \sigma_{X_1}} & \ldots & \frac{\mathrm{E}\left[(X_1-\mathrm{E}[X_1]) (X_n-\mathrm{E}[X_n])\right]}{\sigma_{X_1} \, \sigma_{X_n}} \\ \vdots & \ddots & \vdots \\ \frac{\mathrm{E}\left[(X_n-\mathrm{E}[X_n]) (X_1-\mathrm{E}[X_1])\right]}{\sigma_{X_n} \, \sigma_{X_1}} & \ldots & \frac{\mathrm{E}\left[(X_n-\mathrm{E}[X_n]) (X_n-\mathrm{E}[X_n])\right]}{\sigma_{X_n} \, \sigma_{X_n}} \end{bmatrix} \; .\]
 
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Metadata: ID: D73 | shortcut: corrmat | author: JoramSoch | date: 2020-06-06, 04:56.