Probability density function of the matrix-normal distribution

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

Theorem: Let $X$ be a random matrix following a matrix-normal distribution:

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

Then, the probability density function of $X$ is

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

Proof: This follows directly from the definition of the matrix-normal distribution.

∎

Sources:

original work

Metadata: ID: P70 | shortcut: matn-pdf | author: JoramSoch | date: 2020-03-02, 21:03.