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

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

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

Then, the mean or expected value of $X$ is

\[\label{eq:matn-mean} \mathrm{E}(X) = M \; .\]

Proof: When $X$ follows a matrix-normal distribution, its vectorized version follows a multivariate normal distribution

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

and the expected value of this multivariate normal distribution is

\[\label{eq:mvn-mean} \mathrm{E}[\mathrm{vec}(X)] = \mathrm{vec}(M) \; .\]

Since the expected value of a random matrix is calculated element-wise, we can invert the vectorization operator to get:

\[\label{eq:matn-mean-qed} \mathrm{E}[X] = M \; .\]

Metadata: ID: P341 | shortcut: matn-mean | author: JoramSoch | date: 2022-09-15, 12:05.