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| 1 | +--- |
| 2 | +layout: proof |
| 3 | +mathjax: true |
| 4 | + |
| 5 | +author: "Joram Soch" |
| 6 | +affiliation: "BCCN Berlin" |
| 7 | +e_mail: "joram.soch@bccn-berlin.de" |
| 8 | +date: 2021-12-07 08:43:00 |
| 9 | + |
| 10 | +title: "Sampling from the matrix-normal distribution" |
| 11 | +chapter: "Probability Distributions" |
| 12 | +section: "Matrix-variate continuous distributions" |
| 13 | +topic: "Matrix-normal distribution" |
| 14 | +theorem: "Linear transformation" |
| 15 | + |
| 16 | +sources: |
| 17 | + - authors: "Wikipedia" |
| 18 | + year: 2021 |
| 19 | + title: "Matrix normal distribution" |
| 20 | + in: "Wikipedia, the free encyclopedia" |
| 21 | + pages: "retrieved on 2021-12-07" |
| 22 | + url: "https://en.wikipedia.org/wiki/Matrix_normal_distribution#Drawing_values_from_the_distribution" |
| 23 | + |
| 24 | +proof_id: "P297" |
| 25 | +shortcut: "matn-samp" |
| 26 | +username: "JoramSoch" |
| 27 | +--- |
| 28 | + |
| 29 | + |
| 30 | +**Theorem:** Let $X \in \mathbb{R}^{n \times p}$ be a [random matrix](/D/rmat) with all entries independently following a [standard normal distribution](/D/snorm). Moreover, let $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{p \times p}$, such that $A A^\mathrm{T} = U$ and $B^\mathrm{T} B = V$. Then, $Y = M + A X B$ follows a [matrix-normal distribution](/D/matn) with [mean](/D/mean-rmat) $M$, [covariance](/D/covmat) across rows $U$ and [covariance](/D/covmat) across rows $U$: |
| 31 | + |
| 32 | +$$ \label{eq:matn-samp} |
| 33 | +Y = M + A X B \sim \mathcal{MN}(M, U, V) \; . |
| 34 | +$$ |
| 35 | + |
| 36 | + |
| 37 | +**Proof:** If all entries of $X$ are independent and [standard normally distributed](/D/snorm) |
| 38 | + |
| 39 | +$$ \label{eq:xij-dist} |
| 40 | +x_{ij} \sim \mathcal{N}(0, 1) \quad \text{ind. for all} \quad i = 1,\ldots,n \quad \text{and} \quad j = 1,\ldots,p \; , |
| 41 | +$$ |
| 42 | + |
| 43 | +this [implies a multivariate normal distribution with diagonal covariance matrix](/P/mvn-ind): |
| 44 | + |
| 45 | +$$ \label{eq:vecX-dist} |
| 46 | +\begin{split} |
| 47 | +\mathrm{vec}(X) &\sim \mathcal{N}\left(\mathrm{vec}(0_{np}), I_{np} \right) \\ |
| 48 | +&\sim \mathcal{N}\left(\mathrm{vec}(0_{np}), I_p \otimes I_n \right) \; . |
| 49 | +\end{split} |
| 50 | +$$ |
| 51 | + |
| 52 | +where $0_{np}$ is an $n \times p$ matrix of zeros and $I_n$ is the $n \times n$ identity matrix. |
| 53 | + |
| 54 | +Due to the [relationship between multivariate and matrix-normal distribution](/P/matn-mvn), we have: |
| 55 | + |
| 56 | +$$ \label{eq:X-dist} |
| 57 | +X \sim \mathcal{MN}(0_{np}, I_n, I_p) \; . |
| 58 | +$$ |
| 59 | + |
| 60 | +Thus, [with the linear transformation theorem for the matrix-normal distribution](/P/matn-ltt), it follows that |
| 61 | + |
| 62 | +$$ \label{eq:matn-samp-qed} |
| 63 | +\begin{split} |
| 64 | +Y = M + AXB &\sim \mathcal{MN}\left(M + A 0_{np} B, A I_n A^\mathrm{T}, B^\mathrm{T} I_p B \right) \\ |
| 65 | +&\sim \mathcal{MN}\left(M, A A^\mathrm{T}, B^\mathrm{T} B \right) \\ |
| 66 | +&\sim \mathcal{MN}\left(M, U, V \right) \; . |
| 67 | +\end{split} |
| 68 | +$$ |
| 69 | + |
| 70 | +Thus, given $X$ defined by \eqref{eq:xij-dist}, $Y$ defined by \eqref{eq:matn-samp} is a [sample](/D/dist) from $\mathcal{N}\left(M, U, V \right)$. |
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