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

Theorem: Let $X$ be an $n \times p$ random matrix following a matrix-normal distribution:

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

Then,

1) the covariance matrix of each row of $X$ is a scalar multiple of $V$

\[\label{eq:matn-cov-row} \mathrm{Cov}(x_{i,\bullet}^\mathrm{T}) \propto V \quad \text{for all} \quad i = 1,\ldots,n \; ;\]

2) the covariance matrix of each column of $X$ is a scalar multiple of $U$

\[\label{eq:matn-cov-col} \mathrm{Cov}(x_{\bullet,j}) \propto U \quad \text{for all} \quad i = 1,\ldots,p \; .\]

Proof:

1) The marginal distribution of a given row of $X$ is a multivariate normal distribution

\[\label{eq:matn-marg-row} x_{i,\bullet}^\mathrm{T} \sim \mathcal{N}(m_{i,\bullet}^\mathrm{T}, u_{ii} V) \; ,\]

and the covariance of this multivariate normal distribution is

\[\label{eq:matn-cov-row-qed} \mathrm{Cov}(x_{i,\bullet}^\mathrm{T}) = u_{ii} V \propto V \; .\]

2) The marginal distribution of a given column of $X$ is a multivariate normal distribution

\[\label{eq:matn-marg-col} x_{\bullet,j} \sim \mathcal{N}(m_{\bullet,j}, v_{jj} U) \; ,\]

and the covariance of this multivariate normal distribution is

\[\label{eq:matn-cov-col-qed} \mathrm{Cov}(x_{\bullet,j}) = v_{jj} U \propto U \; .\]
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Metadata: ID: P342 | shortcut: matn-cov | author: JoramSoch | date: 2022-09-15, 12:23.