Proof: Redundancy of parameters describing the matrix-normal distribution
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: The probability density function of the matrix-normal distribution is
\[\label{eq:matn} \begin{split} X &\sim \mathcal{MN}(M, U, V) \\ \Rightarrow \quad f(X) &= \frac{1}{\sqrt{(2\pi)^{np} |V|^n |U|^p}} \cdot \exp\left[-\frac{1}{2} \mathrm{tr}\left( V^{-1} (X-M)^\mathrm{T} \, U^{-1} (X-M) \right) \right] \; . \end{split}\]Using the inverse matrix property $(cA)^{-1} = (1/c) A^{-1}$ and the matrix determinant property $\lvert cA \rvert = c^n \lvert A \rvert$, the probability density function of the second distribution in \eqref{eq:matn-red} becomes
\[\label{eq:matn-red-qed} \begin{split} p(X) &= \frac{1}{\sqrt{(2\pi)^{np} \left| \frac{1}{a} V \right|^n \left| a U \right|^p}} \cdot \exp\left[-\frac{1}{2} \mathrm{tr}\left( \left( \frac{1}{a} V \right)^{-1} (X-M)^\mathrm{T} \, \left( a U \right)^{-1} (X-M) \right) \right] \\ &= \frac{1}{\sqrt{(2\pi)^{np} \left( \left( \frac{1}{a} \right)^p |V| \right)^n \left( a^n |U| \right)^p}} \cdot \exp\left[-\frac{1}{2} \mathrm{tr}\left( \frac{a}{a} V^{-1} (X-M)^\mathrm{T} \, U^{-1} (X-M) \right) \right] \\ &= \frac{1}{\sqrt{(2\pi)^{np} \left( \frac{1}{a} \right)^{np} a^{np} |V|^n |U|^p}} \cdot \exp\left[-\frac{1}{2} \mathrm{tr}\left( V^{-1} (X-M)^\mathrm{T} \, U^{-1} (X-M) \right) \right] \\ &= \frac{1}{\sqrt{(2\pi)^{np} |V|^n |U|^p}} \cdot \exp\left[-\frac{1}{2} \mathrm{tr}\left( V^{-1} (X-M)^\mathrm{T} \, U^{-1} (X-M) \right) \right] \end{split}\]which is equal to the probability density function of the first distribution in \eqref{eq:matn-red}.
- Glanz, Hunter; Carvalho, Luis (2013): "An Expectation-Maximization Algorithm for the Matrix Normal Distribution"; in: arXiv stat.ME, sect. 2.1, p. 3; URL: https://arxiv.org/abs/1309.6609; DOI: 10.48550/arXiv.1309.6609.
Metadata: ID: P506 | shortcut: matn-red2 | author: JoramSoch | date: 2025-06-24, 12:21.