Index: The Book of Statistical ProofsProbability DistributionsMatrix-variate continuous distributionsMatrix-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: 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}.

Sources:

Metadata: ID: P506 | shortcut: matn-red2 | author: JoramSoch | date: 2025-06-24, 12:21.