Proof: Cross-covariance matrices of the matrix-normal distribution
Theorem: Let $X$ be an $n \times p$ random matrix following a matrix-normal distribution:
\[\label{eq:matn} X \sim \mathcal{MN}(M, U, V) \; .\]Then,
1) the cross-covariance matrix of any two columns of $X$ is equal to $U$, multiplied with the corresponding entry of $V$:
\[\label{eq:matn-ccm-col} \mathrm{Cov}(x_{\bullet i}, x_{\bullet j}) = v_{ij} \, U \quad \text{where} \quad i,j = 1,\ldots,p \; ;\]2) the cross-covariance matrix of any two rows of $X$ is equal to $V$, multiplied with the corresponding entry of $U$:
\[\label{eq:matn-ccm-row} \mathrm{Cov}(x_{i \bullet}^\mathrm{T}, x_{j \bullet}^\mathrm{T}) = u_{ij} \, V \quad \text{where} \quad i,j = 1,\ldots,n \; ;\]3) the cross-covariance matrix of a given row and column from $X$ is equal to the matrix product of the corresponding row and column from $V$ and $U$:
\[\label{eq:matn-ccm-row-col} \mathrm{Cov}(x_{i \bullet}^\mathrm{T}, x_{\bullet j}) = v_{\bullet j} \, u_{i \bullet} \quad \text{where} \quad i = 1,\ldots,n, \; j = 1,\ldots,p \; .\]Proof: When the matrix $X \in \mathbb{R}^{n \times p}$ follows a matrix-normal distribution, the vector $\mathrm{vec}(X) \in \mathbb{R}^{np}$ follows a multivariate normal distribution:
\[\label{eq:matn-mvn} \mathrm{vec}(X) \sim \mathcal{N}(\mathrm{vec}(M), V \otimes U)\]where $\mathrm{vec}(A)$ is the vectorization function which collects the entries of an $n \times m$ matrix $A$ into a single $(n \cdot m)$-dimensional vector column by column:
\[\label{eq:vec} \mathrm{vec}(A) = \left[ \begin{matrix} x_{11} \\ \vdots \\ x_{n1} \\ \\ \vdots \\ \\ x_{1m} \\ \vdots \\ x_{nm} \end{matrix} \right] \; .\]Any marginal distribution of a multivariate normal distribution can be obtained by removing the corresponding entries (i.e. the ones marginalized out) from mean vector and covariance matrix:
\[\label{eq:mvn-marg} X \sim \mathcal{N}(\mu, \Sigma) \quad \Rightarrow \quad X_s \sim \mathcal{N}(\mu_s, \Sigma_s) \; .\]Moreover, the covariance matrix of a multivariate normal distribution is equal to the second parameter of the distribution:
\[\label{eq:mvn-cov} X \sim \mathcal{N}(\mu, \Sigma) \quad \Rightarrow \quad \mathrm{Cov}(X) = \Sigma \; .\]Thus, the above cross-covariance matrices can be read out from the non-diagonal entries of covariance matrices belonging to marginal distribution of corresponding rows and columns according to \eqref{eq:matn-mvn} and \eqref{eq:mvn-marg}.
1) The marginal distribution of two columns $x_{\bullet i}$ and $x_{\bullet j}$ is
\[\label{eq:matn-marg-col} \left[ \begin{matrix} x_{\bullet i} \\ x_{\bullet j} \end{matrix} \right] \sim \mathcal{N} \left( \left[ \begin{matrix} m_{\bullet i} \\ m_{\bullet j} \end{matrix} \right], \left[ \begin{matrix} v_{ii} & v_{ij} \\ v_{ji} & v_{jj} \end{matrix} \right] \otimes U \right) \; .\]Thus, the cross-covariance of $x_{\bullet i}$ and $x_{\bullet j}$ is
\[\label{eq:matn-ccm-col-qed} \mathrm{Cov}(x_{\bullet i}, x_{\bullet j}) = v_{ij} \, U \; .\]2) The marginal distribution of two rows $x_{i \bullet}$ and $x_{j \bullet}$ is
\[\label{eq:matn-marg-row} \left[ \begin{matrix} x_{i \bullet}^\mathrm{T} \\ x_{j \bullet}^\mathrm{T} \end{matrix} \right] \sim \mathcal{N} \left( \left[ \begin{matrix} m_{i \bullet}^\mathrm{T} \\ m_{j \bullet}^\mathrm{T} \end{matrix} \right], V \otimes \left[ \begin{matrix} u_{ii} & u_{ij} \\ u_{ji} & u_{jj} \end{matrix} \right] \right) \; .\]Thus, the cross-covariance of $x_{i \bullet}$ and $x_{j \bullet}$ is
\[\label{eq:matn-ccm-row-qed} \mathrm{Cov}(x_{i \bullet}^\mathrm{T}, x_{j \bullet}^\mathrm{T}) = u_{ij} \, V \; .\]3) The cross-covariance matrix of a row $x_{i \bullet}$ and a column $x_{\bullet j}$ can be obtained by extracting the rows $i, n+i, 2n+i, \ldots$ (i.e. belonging to $x_{i \bullet}$) and the columns $j, p+j, 2p+j, \ldots$ (i.e. belonging to $x_{\bullet j}$) from the matrix $V \otimes U$.
Thus, the cross-covariance of $x_{i \bullet}$ and $x_{\bullet j}$ is
\[\label{eq:matn-ccm-row-col-qed} \mathrm{Cov}(x_{i \bullet}^\mathrm{T}, x_{\bullet j}) = v_{\bullet j} \, u_{i \bullet} \; .\]Note that $v_{\bullet j} \in \mathbb{R}^{p \times 1}$ and $u_{i \bullet} \in \mathbb{R}^{1 \times n}$, such that $v_{\bullet j} \, u_{i \bullet} \in \mathbb{R}^{p \times n}$, compatible with $x_{i \bullet}^\mathrm{T} \in \mathbb{R}^p$ ($p$ entries per row) and $x_{\bullet j} \in \mathbb{R}^n$ ($n$ entries per column).
Metadata: ID: P522 | shortcut: matn-ccm | author: JoramSoch | date: 2026-01-23, 11:07.