Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Matrix-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.

Proof: The matrix-normal distribution is equivalent to the multivariate normal distribution,

\[\label{eq:matn-mvn} X \sim \mathcal{MN}(M, U, V) \quad \Leftrightarrow \quad \mathrm{vec}(X) \sim \mathcal{N}(\mathrm{vec}(M), V \otimes U) \; ,\]

and the linear transformation theorem for the multivariate normal distribution states:

\[\label{eq:mvn-ltt} x \sim \mathcal{N}(\mu, \Sigma) \quad \Rightarrow \quad y = Ax + b \sim \mathcal{N}(A\mu + b, A \Sigma A^\mathrm{T}) \; .\]

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:

\[\label{eq:matn-ltt-qed} Y \sim \mathcal{MN}(AMB + C, AUA^\mathrm{T} ,B^\mathrm{T}VB) \; .\]
∎
Sources:

Metadata: ID: P145 | shortcut: matn-ltt | author: JoramSoch | date: 2020-08-03, 22:24.