added 2 proofs

Author: JoramSoch

P/matn-samp.md

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+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2021-12-07 08:43:00
+
+title: "Sampling from the matrix-normal distribution"
+chapter: "Probability Distributions"
+section: "Matrix-variate continuous distributions"
+topic: "Matrix-normal distribution"
+theorem: "Linear transformation"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2021
+    title: "Matrix normal distribution"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2021-12-07"
+    url: "https://en.wikipedia.org/wiki/Matrix_normal_distribution#Drawing_values_from_the_distribution"
+
+proof_id: "P297"
+shortcut: "matn-samp"
+username: "JoramSoch"
+---
+
+
+**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 $U$:
+
+$$ \label{eq:matn-samp}
+Y = M + A X B \sim \mathcal{MN}(M, U, V) \; .
+$$
+
+
+**Proof:** If all entries of $X$ are independent and [standard normally distributed](/D/snorm)
+
+$$ \label{eq:xij-dist}
+x_{ij} \sim \mathcal{N}(0, 1) \quad \text{ind. for all} \quad i = 1,\ldots,n \quad \text{and} \quad j = 1,\ldots,p \; ,
+$$
+
+this [implies a multivariate normal distribution with diagonal covariance matrix](/P/mvn-ind):
+
+$$ \label{eq:vecX-dist}
+\begin{split}
+\mathrm{vec}(X) &\sim \mathcal{N}\left(\mathrm{vec}(0_{np}), I_{np} \right) \\
+&\sim \mathcal{N}\left(\mathrm{vec}(0_{np}), I_p \otimes I_n \right) \; .
+\end{split}
+$$
+
+where $0_{np}$ is an $n \times p$ matrix of zeros and $I_n$ is the $n \times n$ identity matrix.
+
+Due to the [relationship between multivariate and matrix-normal distribution](/P/matn-mvn), we have:
+
+$$ \label{eq:X-dist}
+X \sim \mathcal{MN}(0_{np}, I_n, I_p) \; .
+$$
+
+Thus, [with the linear transformation theorem for the matrix-normal distribution](/P/matn-ltt), it follows that
+
+$$ \label{eq:matn-samp-qed}
+\begin{split}
+Y = M + AXB &\sim \mathcal{MN}\left(M + A 0_{np} B, A I_n A^\mathrm{T}, B^\mathrm{T} I_p B \right) \\
+&\sim \mathcal{MN}\left(M, A A^\mathrm{T}, B^\mathrm{T} B \right) \\
+&\sim \mathcal{MN}\left(M, U, V \right) \; .
+\end{split}
+$$
+
+Thus, given $X$ defined by \eqref{eq:xij-dist}, $Y$ defined by \eqref{eq:matn-samp} is a [sample](/D/dist) from $\mathcal{N}\left(M, U, V \right)$.

P/mean-tr.md

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+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2021-12-07 09:03:00
+
+title: "Expected value of the trace of a matrix"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Expected value"
+theorem: "Expectation of a trace"
+
+sources:
+  - authors: "drerD"
+    year: 2018
+    title: "'Trace trick' for expectations of quadratic forms"
+    in: "StackExchange Mathematics"
+    pages: "retrieved on 2021-12-07"
+    url: "https://math.stackexchange.com/a/3004034/480910"
+
+proof_id: "P298"
+shortcut: "mean-tr"
+username: "JoramSoch"
+---
+
+
+**Theorem:** Let $A$ be an $n \times n$ [random matrix](/D/rmat). Then, the [expected value](/D/mean) of the trace of $A$ is equal to the trace of the [expectation](/D/mean) of $A$:
+
+$$ \label{eq:mean-tr}
+\mathrm{E}\left[ \mathrm{tr}(A) \right] = \mathrm{tr}\left( \mathrm{E}[A] \right) \; .
+$$
+
+
+**Proof:** The trace of an $n \times n$ matrix $A$ is defined as:
+
+$$ \label{eq:tr}
+\mathrm{tr}(A) = \sum_{i=1}^{n} a_{ii} \; .
+$$
+
+Using this definition of the trace, the [linearity of the expected value](/P/mean-lin) and the [expected value of a random matrix](/D/mean-rmat), we have:
+
+$$ \label{eq:mean-tr-qed}
+\begin{split}
+\mathrm{E}\left[ \mathrm{tr}(A) \right] &= \mathrm{E}\left[ \sum_{i=1}^{n} a_{ii} \right] \\
+&= \sum_{i=1}^{n} \mathrm{E}\left[ a_{ii} \right] \\
+&= \mathrm{tr}\left( \left[ \begin{matrix} \mathrm{E}[a_{11}] & \ldots & \mathrm{E}[a_{1n}] \\ \vdots & \ddots & \vdots \\ \mathrm{E}[a_{n1}] & \ldots & \mathrm{E}[a_{nn}] \end{matrix} \right] \right) \\
+&= \mathrm{tr}\left( \mathrm{E}[A] \right) \; .
+\end{split}
+$$