added 1 definition

Author: JoramSoch

D/covmat-cross.md

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+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2022-09-26 09:45:00
+
+title: "Cross-covariance matrix"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Covariance"
+definition: "Cross-covariance matrix"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2022
+    title: "Cross-covariance matrix"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2022-09-26"
+    url: "https://en.wikipedia.org/wiki/Cross-covariance_matrix#Definition"
+
+def_id: "D176"
+shortcut: "covmat-cross"
+username: "JoramSoch"
+---
+
+
+**Definition:** Let $X = [X_1, \ldots, X_n]^\mathrm{T}$ and $Y = [Y_1, \ldots, Y_m]^\mathrm{T}$ be two [random vectors](/D/rvec) that can or cannot be of equal size. Then, the cross-covariance matrix of $X$ and $Y$ is defined as the $n \times m$ matrix in which the entry $(i,j)$ is the [covariance](/D/cov) of $X_i$ and $Y_j$:
+
+$$ \label{eq:covmat-cross}
+\Sigma_{XY} =
+\begin{bmatrix}
+\mathrm{Cov}(X_1,Y_1) & \ldots & \mathrm{Cov}(X_1,Y_m) \\
+\vdots & \ddots & \vdots \\
+\mathrm{Cov}(X_n,Y_1) & \ldots & \mathrm{Cov}(X_n,Y_m)
+\end{bmatrix} =
+\begin{bmatrix}
+\mathrm{E}\left[ (X_1-\mathrm{E}[X_1]) (Y_1-\mathrm{E}[Y_1]) \right] & \ldots & \mathrm{E}\left[ (X_1-\mathrm{E}[X_1]) (Y_m-\mathrm{E}[Y_m]) \right] \\
+\vdots & \ddots & \vdots \\
+\mathrm{E}\left[ (X_n-\mathrm{E}[X_n]) (Y_1-\mathrm{E}[Y_1]) \right] & \ldots & \mathrm{E}\left[ (X_n-\mathrm{E}[X_n]) (Y_m-\mathrm{E}[Y_m]) \right]
+\end{bmatrix} \; .
+$$