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

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, quadratic forms of $X$ have the following expectations:

1) for a $p \times p$ matrix $A$, we have

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

2) for an $n \times n$ matrix $B$, we have

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

3) for a $p \times n$ matrix $C$, we have

\[\label{eq:matn-mean-XCX} \mathrm{E}\left[ X C X \right] = M C M + U C^\mathrm{T} V \; .\]

Proof: The expected value of a random matrix is equal to the matrix of expected values:

\[\label{eq:mean-rmat} \mathrm{E}\left( \left[ \begin{array}{ccc} X_{11} & \ldots & X_{1p} \\ \vdots & \ddots & \vdots \\ X_{n1} & \ldots & X_{np} \end{array} \right] \right) = \left[ \begin{array}{ccc} \mathrm{E}(X_{11}) & \ldots & \mathrm{E}(X_{1p}) \\ \vdots & \ddots & \vdots \\ \mathrm{E}(X_{n1}) & \ldots & \mathrm{E}(X_{np}) \end{array} \right] \; .\]

The expectation of a bilinear form $X^\mathrm{T} A Y$ with $X \in \mathbb{R}^{n}$, $Y \in \mathbb{R}^{m}$ and $A \in \mathbb{R}^{n \times m}$ is equal to

\[\label{eq:mean-blf} \mathrm{E}\left[ X^\mathrm{T} A Y \right] = \mu_X^\mathrm{T} A \mu_Y + \mathrm{tr}(A^\mathrm{T} \Sigma_{XY})\]

where $\mu_X = \mathrm{E}(X)$, $\mu_Y = \mathrm{E}(Y)$ and $\Sigma_{XY}$ is the $n \times m$ cross-covariance matrix of $X$ and $Y$.

When the matrix $X \in \mathbb{R}^{n \times p}$ follows a matrix-normal distribution, we have the following cross-covariance matrices of rows and columns from $X$:

\[\label{eq:matn-ccm-col} \mathrm{Cov}(x_{\bullet i}, x_{\bullet j}) = v_{ij} \, U \quad \text{where} \quad i,j = 1,\ldots,p \; ;\] \[\label{eq:matn-ccm-row} \mathrm{Cov}(x_{i \bullet}^\mathrm{T}, x_{j \bullet}^\mathrm{T}) = u_{ij} \, V \quad \text{where} \quad i,j = 1,\ldots,n \; ;\] \[\label{eq:matn-ccm-row-col} \mathrm{Cov}(x_{i \bullet}^\mathrm{T}, x_{\bullet j}) = v_{\bullet j} \, u_{i \bullet} \quad \text{where} \quad i = 1,\ldots,n, \; j = 1,\ldots,p \; .\]

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

1) The $(i,j)$-th entry of the expectation $\mathrm{E}\left[ X A X^\mathrm{T} \right]$ is

\[\label{eq:matn-mean-XAXT-qed} \begin{split} \left( \mathrm{E}\left[ X A X^\mathrm{T} \right] \right)_{ij} &\overset{\eqref{eq:mean-rmat}}{=} \mathrm{E}\left[ \left( X A X^\mathrm{T} \right)_{ij} \right] \\ &= \mathrm{E}\left[ x_{i \bullet} A x_{j \bullet}^\mathrm{T} \right] \\ &\overset{\eqref{eq:mean-blf}}{=} m_{i \bullet} A m_{j \bullet}^\mathrm{T} + \mathrm{tr}(A^\mathrm{T} \mathrm{Cov}[x_{i \bullet}^\mathrm{T}, x_{j \bullet}^\mathrm{T}]) \\ &\overset{\eqref{eq:matn-ccm-row}}{=} m_{i \bullet} A m_{j \bullet}^\mathrm{T} + \mathrm{tr}(A^\mathrm{T} u_{ij} V) \\ &= m_{i \bullet} A m_{j \bullet}^\mathrm{T} + u_{ij} \, \mathrm{tr}(A^\mathrm{T} V) \\ \Rightarrow \mathrm{E}\left[ X A X^\mathrm{T} \right] &= M A M^\mathrm{T} + U \, \mathrm{tr}(A^\mathrm{T} V) \; . \end{split}\]

2) The $(i,j)$-th entry of the expectation $\mathrm{E}\left[ X^\mathrm{T} B X \right]$ is

\[\label{eq:matn-mean-XTBX-qed} \begin{split} \left( \mathrm{E}\left[ X^\mathrm{T} B X \right] \right)_{ij} &\overset{\eqref{eq:mean-rmat}}{=} \mathrm{E}\left[ \left( X B X^\mathrm{T} \right)_{ij} \right] \\ &= \mathrm{E}\left[ x_{\bullet i}^\mathrm{T} B x_{\bullet j} \right] \\ &\overset{\eqref{eq:mean-blf}}{=} m_{\bullet i}^\mathrm{T} B m_{\bullet j} + \mathrm{tr}(B^\mathrm{T} \mathrm{Cov}[x_{\bullet i}^\mathrm{T}, x_{\bullet j}]) \\ &\overset{\eqref{eq:matn-ccm-col}}{=} m_{\bullet i}^\mathrm{T} B m_{\bullet j} + \mathrm{tr}(B^\mathrm{T} v_{ij} U) \\ &= m_{\bullet i}^\mathrm{T} B m_{\bullet j} + v_{ij} \, \mathrm{tr}(U B^\mathrm{T}) \\ \Rightarrow \mathrm{E}\left[ X^\mathrm{T} B X \right] &= M^\mathrm{T} B M + V \, \mathrm{tr}(U B^\mathrm{T}) \; . \end{split}\]

3) The $(i,j)$-th entry of the expectation $\mathrm{E}\left[ X C X \right]$ is

\[\label{eq:matn-mean-XCX-qed} \begin{split} \left( \mathrm{E}\left[ X C X \right] \right)_{ij} &\overset{\eqref{eq:mean-rmat}}{=} \mathrm{E}\left[ \left( X C X \right)_{ij} \right] \\ &= \mathrm{E}\left[ x_{i \bullet} C x_{\bullet j} \right] \\ &\overset{\eqref{eq:mean-blf}}{=} m_{i \bullet} C m_{\bullet j} + \mathrm{tr}(C^\mathrm{T} \mathrm{Cov}[x_{i \bullet}^\mathrm{T}, x_{\bullet j}]) \\ &\overset{\eqref{eq:matn-ccm-row-col}}{=} m_{i \bullet} C m_{\bullet j} + \mathrm{tr}(C^\mathrm{T} v_{\bullet j} u_{i \bullet}) \\ &= m_{i \bullet} C m_{\bullet j} + \mathrm{tr}(u_{i \bullet} C^\mathrm{T} v_{\bullet j}) \\ &= m_{i \bullet} C m_{\bullet j} + u_{i \bullet} C^\mathrm{T} v_{\bullet j} \\ \Rightarrow \mathrm{E}\left[ X C X \right] &= M C M + U C^\mathrm{T} V \; . \end{split}\]
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Metadata: ID: P523 | shortcut: matn-meanqf | author: JoramSoch | date: 2026-01-23, 12:01.