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

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, we have the following second-order expectations for $X$:

1) the second-order expectation across rows is

\[\label{eq:matn-mean-XMXMT} \mathrm{E}\left[ (X-M) (X-M)^\mathrm{T} \right] = U \, \mathrm{tr}(V) \; ;\]

2) the second-order expectation across columns is

\[\label{eq:matn-mean-XMTXM} \mathrm{E}\left[ (X-M)^\mathrm{T} (X-M) \right] = V \, \mathrm{tr}(U) \; .\]

Proof: The linear transformation theorem for the matrix-normal distribution states that any linear transformation of a normal random matrix is again matrix-normally distributed:

\[\label{eq:matn-ltt} X \sim \mathcal{MN}(M, U, V) \quad \Rightarrow \quad Y = AXB + C \sim \mathcal{MN}(AMB+C, AUA^\mathrm{T}, B^\mathrm{T}VB) \; .\]

Applying \eqref{eq:matn-ltt} to \eqref{eq:matn}, we get:

\[\label{eq:XM-dist} \begin{split} Y = X - M = I_n X \, I_p - M &\sim \mathcal{MN}(I_n M I_p - M, I_n U I_n^\mathrm{T}, I_p^\mathrm{T} V I_p) \\ &\sim \mathcal{MN}(0_{np}, U, V) \; . \end{split}\]

Moreover, when the matrix $X \in \mathbb{R}^{n \times p}$ follows a matrix-normal distribution, we have the following expectations of quadratic forms for $X$:

\[\label{eq:matn-mean-XAXT} \mathrm{E}\left[ X A X^\mathrm{T} \right] = M A M^\mathrm{T} + U \, \mathrm{tr}(A^\mathrm{T} V) \; ;\] \[\label{eq:matn-mean-XTBX} \mathrm{E}\left[ X^\mathrm{T} B X \right] = M^\mathrm{T} B M + V \, \mathrm{tr}(U B^\mathrm{T}) \; ;\] \[\label{eq:matn-mean-XCX} \mathrm{E}\left[ X C X \right] = M C M + U C^\mathrm{T} V \; .\]

With that, we are able to derive the above equations:

1) The second-order expectation across rows obtains as

\[\label{eq:matn-mean-XMXMT-qed} \begin{split} \mathrm{E}\left[ (X-M) (X-M)^\mathrm{T} \right] &\overset{\eqref{eq:XM-dist}}{=} \mathrm{E}\left[ Y Y^\mathrm{T} \right] \\ &= \mathrm{E}\left[ Y I_p Y^\mathrm{T} \right] \\ &\overset{\eqref{eq:matn-mean-XAXT}}{=} 0_{np} I_p 0_{np}^\mathrm{T} + U \, \mathrm{tr}(I_p^\mathrm{T} V) \\ &= U \, \mathrm{tr}(V) \; . \end{split}\]

2) The second-order expectation across columns obtains as

\[\label{eq:matn-mean-XMTXM-qed} \begin{split} \mathrm{E}\left[ (X-M)^\mathrm{T} (X-M) \right] &\overset{\eqref{eq:XM-dist}}{=} \mathrm{E}\left[ Y^\mathrm{T} Y \right] \\ &= \mathrm{E}\left[ Y^\mathrm{T} I_n Y \right] \\ &\overset{\eqref{eq:matn-mean-XTBX}}{=} 0_{np}^\mathrm{T} I_n 0_{np} + V \, \mathrm{tr}(I_n^\mathrm{T} U) \\ &= V \, \mathrm{tr}(U) \; . \end{split}\]
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Metadata: ID: P524 | shortcut: matn-meanso | author: JoramSoch | date: 2026-01-23, 13:49.