matn-mean.md

layout	mathjax	author	affiliation	e_mail	date	title	chapter	section	topic	theorem	sources	proof_id	shortcut	username
proof	true	Joram Soch	BCCN Berlin	joram.soch@bccn-berlin.de	2022-09-15 05:05:00 -0700	Mean of the matrix-normal distribution	Probability Distributions	Matrix-variate continuous distributions	Matrix-normal distribution	Mean	authors year title in pages url Wikipedia 2022 Matrix normal distribution Wikipedia, the free encyclopedia retrieved on 2022-09-15 https://en.wikipedia.org/wiki/Matrix_normal_distribution#Expected_values	P341	matn-mean	JoramSoch

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:

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