Index: The Book of Statistical ProofsProbability Distributions ▷ Matrix-variate continuous distributions ▷ Matrix-normal distribution ▷ Definition

Definition: Let $X$ be an $n \times p$ random matrix. Then, $X$ is said to be matrix-normally distributed with mean $M$, covariance across rows $U$ and covariance across columns $V$

\[\label{eq:matn} X \sim \mathcal{MN}(M, U, V) \; ,\]

if and only if its probability density function is given by

\[\label{eq:matn-pdf} \mathcal{MN}(X; M, U, V) = \frac{1}{\sqrt{(2\pi)^{np} |V|^n |U|^p}} \cdot \exp\left[-\frac{1}{2} \mathrm{tr}\left( V^{-1} (X-M)^\mathrm{T} \, U^{-1} (X-M) \right) \right]\]

where $M$ is an $n \times p$ real matrix, $U$ is an $n \times n$ positive definite matrix and $V$ is a $p \times p$ positive definite matrix.


Metadata: ID: D6 | shortcut: matn | author: JoramSoch | date: 2020-01-27, 14:37.