corrected some pages

Several small mistakes/errors were corrected in several proofs/definitions.

Author: JoramSoch

P/matn-marg.md

@@ -148,7 +148,7 @@ $$ \label{eq:xj-marg-mvn}
 x_{\bullet,j} \sim \mathcal{N}(m_{\bullet,j}, v_{jj} U) \; .
 $$
 
-4) This is a special case of 1). Setting $A$ to the $i$-th elementary row vector in $n$ dimensions and $B$ to the $j$-th elementary row vector in $p$ dimensions
+4) This is a special case of 2) and 3). Setting $A$ to the $i$-th elementary row vector in $n$ dimensions and $B$ to the $j$-th elementary row vector in $p$ dimensions
 
 $$ \label{eq:AB-elem}
 A = e_i, \; B = e_j^\mathrm{T} \; ,

P/matn-samp.md

@@ -27,7 +27,9 @@ 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 $U$:
+**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 $U$:
 
 $$ \label{eq:matn-samp}
 Y = M + A X B \sim \mathcal{MN}(M, U, V) \; .
@@ -45,7 +47,7 @@ this [implies a multivariate normal distribution with diagonal covariance matrix
 $$ \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) \; .
+&\sim \mathcal{N}\left(\mathrm{vec}(0_{np}), I_p \otimes I_n \right)
 \end{split}
 $$
 
@@ -67,4 +69,4 @@ Y = M + AXB &\sim \mathcal{MN}\left(M + A 0_{np} B, A I_n A^\mathrm{T}, B^\mathr
 \end{split}
 $$
 
-Thus, given $X$ defined by \eqref{eq:xij-dist}, $Y$ defined by \eqref{eq:matn-samp} is a [sample](/D/samp) from $\mathcal{N}\left(M, U, V \right)$.
+Thus, given $X$ defined by \eqref{eq:xij-dist}, $Y$ defined by \eqref{eq:matn-samp} is a [sample](/D/samp) from $\mathcal{MN}\left(M, U, V \right)$.

P/mvn-cov.md

@@ -84,13 +84,13 @@ $$ \label{eq:x-mean}
 \mathrm{Cov}(x) = \mathrm{Cov}( Az + \mu ) \; .
 $$
 
-With the [invariance of the covariance matrix under addition](/P/cov-inv)
+With the [invariance of the covariance matrix under addition](/P/covmat-inv)
 
 $$ \label{eq:cov-inv}
 \mathrm{Cov}(x + a) = \mathrm{Cov}(x)
 $$
 
-and the [scaling of the covariance matrix upon multiplication](/P/cov-scal)
+and the [scaling of the covariance matrix upon multiplication](/P/covmat-scal)
 
 $$ \label{eq:cov-scal}
 \mathrm{Cov}(Ax) = A \mathrm{Cov}(x) A^\mathrm{T} \; ,

P/ng-cov.md

@@ -32,19 +32,19 @@ Then,
 1) the [covariance](/D/cov) of $x$, [conditional](/D/dist-cond) on $y$ is
 
 $$ \label{eq:ng-cov-cond}
-\mathrm{Cov}[x|y] = \frac{1}{y} \Lambda^{-1} \; ;
+\mathrm{Cov}(x \vert y) = \frac{1}{y} \Lambda^{-1} \; ;
 $$
 
 2) the [covariance](/D/cov) of $x$, [unconditional](/D/dist-marg) on $y$ is
 
 $$ \label{eq:ng-cov-x}
-\mathrm{Cov}[x] = \frac{b}{a-1} \Lambda^{-1} \; ;
+\mathrm{Cov}(x) = \frac{b}{a-1} \Lambda^{-1} \; ;
 $$
 
 3) the [variance](/D/var) of $y$ is
 
 $$ \label{eq:ng-var-y}
-\mathrm{Var}[y] = \frac{a}{b^2} \; .
+\mathrm{Var}(y) = \frac{a}{b^2} \; .
 $$
 
 
@@ -65,7 +65,7 @@ $$
 such that we have:
 
 $$ \label{eq:ng-cov-cond-qed}
-\mathrm{Cov}[x|y] = (y \Lambda)^{-1} = \frac{1}{y} \Lambda^{-1} \; .
+\mathrm{Cov}(x \vert y) = (y \Lambda)^{-1} = \frac{1}{y} \Lambda^{-1} \; .
 $$
 
 2) The [marginal distribution of the normal-gamma distribution](/P/ng-marg) with respect to $x$ is a [multivariate t-distribution](/D/mvt):
@@ -83,7 +83,7 @@ $$
 such that we have:
 
 $$ \label{eq:ng-cov-x-qed}
-\mathrm{Cov}[x] = \frac{2a}{2a-2} \left(\frac{a}{b} \Lambda \right)^{-1} = \frac{a}{a-1} \, \frac{b}{a} \, \Lambda^{-1} = \frac{b}{a-1} \Lambda^{-1} \; .
+\mathrm{Cov}(x) = \frac{2a}{2a-2} \left(\frac{a}{b} \Lambda \right)^{-1} = \frac{a}{a-1} \, \frac{b}{a} \, \Lambda^{-1} = \frac{b}{a-1} \Lambda^{-1} \; .
 $$
 
 3) The [marginal distribution of the normal-gamma distribution](/P/ng-marg) with respect to $y$ is a [univariate gamma distribution](/D/gam):
@@ -101,5 +101,5 @@ $$
 such that we have:
 
 $$ \label{eq:ng-var-y-qed}
-\mathrm{Var}[y] = \frac{a}{b^2} \; .
+\mathrm{Var}(y) = \frac{a}{b^2} \; .
 $$

P/ng-samp.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Theorem:** Let $Z_1 \in \mathbb{R}^{n \times 1}$ be a [random vector](/D/rvec) with all entries independently following a [standard normal distribution](/D/snorm) and let $Z_2 \in \mathbb{R}$ be a [random variable](/D/rvar) following a [standard gamma distribution](/D/sgam) with shape $a$. Moreover, let $A \in \mathbb{R}^{n \times n}$ be a matrix such that, such that $A A^\mathrm{T} = \Lambda^{-1}$.
+**Theorem:** Let $Z_1 \in \mathbb{R}^n$ be a [random vector](/D/rvec) with all entries independently following a [standard normal distribution](/D/snorm) and let $Z_2 \in \mathbb{R}$ be a [random variable](/D/rvar) following a [standard gamma distribution](/D/sgam) with shape $a$. Moreover, let $A \in \mathbb{R}^{n \times n}$ be a matrix, such that $A A^\mathrm{T} = \Lambda^{-1}$.
 
 Then, $X = \mu + A Z_1 / \sqrt{Z_2/b}$ and $Y = Z_2/b$ jointly follow a [normal-gamma distribution](/D/ng) with [mean vector](/D/mean-rvec) $\mu$, [precision matrix](/D/precmat) $\Lambda$, shape parameter $a$ and rate parameter $b$: