Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Matrix-variate continuous distributions ▷ Matrix-normal distribution ▷ Redundancy of parameters

Theorem: The covariance parameters of the matrix-normal distribution are redundant up to a scalar factor, i.e. the two probability distributions

\[\label{eq:matn-red} \begin{split} X &\sim \mathcal{MN}(M, U, V) \\ X &\sim \mathcal{MN}\left( M, a \cdot U, \frac{1}{a} \cdot V \right) \end{split}\]

are equivalent for any $a \in \mathbb{R}$ with $a > 0$ where $X \in \mathbb{R}^{n \times p}$ is a random matrix and $U \in \mathbb{R}^{n \times n}$ and $V \in \mathbb{R}^{p \times p}$ are positive-definite matrices.

Proof: Since $U$ and $V$ must be positive-definite in the matrix-normal distribution, the scalar $a$ must be larger than zero. A random matrix follows a matrix-normal distribution, if and only if its vectorization is multivariate normally distributed

\[\label{eq:matn-mvn} X \sim \mathcal{MN}(M, U, V) \quad \Leftrightarrow \quad \mathrm{vec}(X) \sim \mathcal{N}(\mathrm{vec}(M), V \otimes U)\]

where $\mathrm{vec}(X)$ is the vectorization operator and $\otimes$ is the Kronecker product. Thus, the second distribution in \eqref{eq:matn-red} is equivalent to

\[\label{eq:matn-red-qed} \begin{split} X \sim \mathcal{MN}\left( M, a \cdot U, \frac{1}{a} \cdot V \right) \quad \Leftrightarrow \quad \mathrm{vec}(X) &\sim \mathcal{N}\left( \mathrm{vec}(M), \frac{1}{a} V \otimes a U \right) \\ &\sim \mathcal{N}\left( \mathrm{vec}(M), \frac{1}{a} \left( V \otimes a U \right) \right) \\ &\sim \mathcal{N}\left( \mathrm{vec}(M), \frac{a}{a} \left( V \otimes U \right) \right) \\ &\sim \mathcal{N}\left( \mathrm{vec}(M), V \otimes U \right) \end{split}\]

which proves the equivalence of the two distributions in \eqref{eq:matn-red}.

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Metadata: ID: P505 | shortcut: matn-red | author: JoramSoch | date: 2025-06-24, 11:55.