Index: The Book of Statistical ProofsProbability DistributionsMatrix-variate continuous distributions ▷ Matrix-normal distribution ▷ Linear transformation

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, a linear transformation of $X$ is also matrix-normally distributed

$\label{eq:matn-trans} Y = AXB + C \sim \mathcal{MN}(AMB+C, AUA^\mathrm{T}, B^\mathrm{T}VB)$

where $A$ us ab $r \times n$ matrix of full rank $r \leq b$ and $B$ is a $p \times s$ matrix of full rank $s \leq p$ and $C$ is an $r \times s$ matrix.

$\label{eq:matn-mvn} X \sim \mathcal{MN}(M, U, V) \quad \Leftrightarrow \quad \mathrm{vec}(X) \sim \mathcal{N}(\mathrm{vec}(M), V \otimes U) \; ,$ $\label{eq:mvn-ltt} x \sim \mathcal{N}(\mu, \Sigma) \quad \Rightarrow \quad y = Ax + b \sim \mathcal{N}(A\mu + b, A \Sigma A^\mathrm{T}) \; .$

The vectorization of $Y = AXB + C$ is

$\label{eq:vec-Y-s1} \begin{split} \mathrm{vec}(Y) &= \mathrm{vec}(AXB + C) \\ &= \mathrm{vec}(AXB) + \mathrm{vec}(C) \\ &= (B^\mathrm{T} \otimes A)\mathrm{vec}(X) + \mathrm{vec}(C) \end{split}$

and the Kronecker product obeys

$\label{eq:kron-prod} (A \otimes B) (C \otimes D) = (AC) \otimes (BD) \; .$

Using \eqref{eq:matn-mvn} and \eqref{eq:mvn-ltt}, we have

$\label{eq:vec-Y-s2} \begin{split} \mathrm{vec}(Y) &\sim \mathcal{N}((B^\mathrm{T} \otimes A) \mathrm{vec}(M) + \mathrm{vec}(C), (B^\mathrm{T} \otimes A) (V \otimes U) (B^\mathrm{T} \otimes A)^\mathrm{T}) \\ &= \mathcal{N}(\mathrm{vec}(AMB) + \mathrm{vec}(C), (B^\mathrm{T}V \otimes AU) (B^\mathrm{T} \otimes A)^\mathrm{T}) \\ &= \mathcal{N}(\mathrm{vec}(AMB + C), B^\mathrm{T}VB \otimes AUA^\mathrm{T}) \; . \end{split}$

Using \eqref{eq:matn-mvn}, we finally have:

$\label{eq:matn-ltt-qed} Y \sim \mathcal{MN}(AMB + C, AUA^\mathrm{T} ,B^\mathrm{T}VB) \; .$
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Metadata: ID: P145 | shortcut: matn-ltt | author: JoramSoch | date: 2020-08-03, 22:24.