Merge pull request #170 from JoramSoch/master

added 1 proof

Author: JoramSoch

I/ToC.md

@@ -440,13 +440,14 @@ title: "Table of Contents"
 
    4.1. Multivariate normal distribution <br>
    &emsp;&ensp; 4.1.1. *[Definition](/D/mvn)* <br>
-   &emsp;&ensp; 4.1.2. **[Probability density function](/P/mvn-pdf)** <br>
-   &emsp;&ensp; 4.1.3. **[Differential entropy](/P/mvn-dent)** <br>
-   &emsp;&ensp; 4.1.4. **[Kullback-Leibler divergence](/P/mvn-kl)** <br>
-   &emsp;&ensp; 4.1.5. **[Linear transformation](/P/mvn-ltt)** <br>
-   &emsp;&ensp; 4.1.6. **[Marginal distributions](/P/mvn-marg)** <br>
-   &emsp;&ensp; 4.1.7. **[Conditional distributions](/P/mvn-cond)** <br>
-   &emsp;&ensp; 4.1.8. **[Conditions for independence](/P/mvn-ind)** <br>
+   &emsp;&ensp; 4.1.2. **[Special case of matrix-normal distribution](/P/mvn-matn)** <br>
+   &emsp;&ensp; 4.1.3. **[Probability density function](/P/mvn-pdf)** <br>
+   &emsp;&ensp; 4.1.4. **[Differential entropy](/P/mvn-dent)** <br>
+   &emsp;&ensp; 4.1.5. **[Kullback-Leibler divergence](/P/mvn-kl)** <br>
+   &emsp;&ensp; 4.1.6. **[Linear transformation](/P/mvn-ltt)** <br>
+   &emsp;&ensp; 4.1.7. **[Marginal distributions](/P/mvn-marg)** <br>
+   &emsp;&ensp; 4.1.8. **[Conditional distributions](/P/mvn-cond)** <br>
+   &emsp;&ensp; 4.1.9. **[Conditions for independence](/P/mvn-ind)** <br>
 
    4.2. Multivariate t-distribution <br>
    &emsp;&ensp; 4.2.1. *[Definition](/D/mvt)* <br>

P/mvn-matn.md

@@ -0,0 +1,42 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2022-07-31 11:00:00
+
+title: "Multivariate normal distribution is a special case of matrix-normal distribution"
+chapter: "Probability Distributions"
+section: "Multivariate continuous distributions"
+topic: "Multivariate normal distribution"
+theorem: "Special case of matrix-normal distribution"
+
+sources:
+
+proof_id: "P330"
+shortcut: "mvn-matn"
+username: "JoramSoch"
+---
+
+
+**Theorem:** The [multivariate normal distribution](/D/mvn) is a special case of the [matrix-normal distribution](/D/matn) with number of variables $p = 1$, i.e. [random matrix](/D/rvar) $X = x \in \mathbb{R}^{n \times 1}$, mean $M = \mu \in \mathbb{R}^{n \times 1}$, covariance across rows $U = \Sigma$ and covariance across columns $V = 1$.
+
+
+**Proof:** The [probability density function of the matrix-normal distribution](/P/matn-pdf) is
+
+$$ \label{eq:matn-pdf}
+\mathcal{MN}(X; M, U, V) = \frac{1}{\sqrt{(2\pi)^{np} |V|^n |U|^p}} \cdot \exp\left[-\frac{1}{2} \mathrm{tr}\left( V^{-1} (X-M)^\mathrm{T} \, U^{-1} (X-M) \right) \right] \; .
+$$
+
+Setting $p = 1$, $X = x$, $M = \mu$, $U = \Sigma$ and $V = 1$, we obtain
+
+\begin{equation} \label{eq:exp-pdf}
+\begin{split}
+\mathcal{MN}(x; \mu, \Sigma, 1) &= \frac{1}{\sqrt{(2\pi)^{n} |1|^n |\Sigma|^1}} \cdot \exp\left[-\frac{1}{2} \mathrm{tr}\left( 1^{-1} (x-\mu)^\mathrm{T} \, \Sigma^{-1} (x-\mu) \right) \right] \\
+&= \frac{1}{\sqrt{(2\pi)^{n} |\Sigma|}} \cdot \exp\left[-\frac{1}{2} (x-\mu)^\mathrm{T} \, \Sigma^{-1} (x-\mu) \right]
+\end{split}
+\end{equation}
+
+which is equivalent to the [probability density function of the multivariate normal distribution](/P/mvn-pdf).