corrected one proof

More required steps were added to the proof "matn-trans" using information from the proof "pdf-invfct".

Author: JoramSoch

P/matn-trans.md

@@ -21,7 +21,7 @@ username: "JoramSoch"
 ---
 
 
-**Theorem:** Let $X$ be a [random matrix](/D/rmat) following a [matrix-normal distribution](/D/matn):
+**Theorem:** Let $X$ be an $n \times p$ [random matrix](/D/rmat) following a [matrix-normal distribution](/D/matn):
 
 $$ \label{eq:matn}
 X \sim \mathcal{MN}(M, U, V) \; .
@@ -34,28 +34,92 @@ X^\mathrm{T} \sim \mathcal{MN}(M^\mathrm{T}, V, U) \; .
 $$
 
 
-**Proof:** The [probability density function of the matrix-normal distribution](/P/matn-pdf) is:
+**Proof:** For a [random vector](/P/rvec) $X \in \mathbb{R}^n$ with [probability density function](/D/pdf) $f_X(x)$, the [probability density function of the invertible function](/P/pdf-invfct) $Y = g(X)$ is
+
+$$ \label{eq:pdf-invfct}
+f_Y(y) = \left\{
+\begin{array}{rl}
+f_X(g^{-1}(y)) \, \left| J_{g^{-1}}(y) \right| \; , & \text{if} \; y \in \mathcal{Y} \\
+0 \; , & \text{if} \; y \notin \mathcal{Y}
+\end{array}
+\right.
+$$
+
+where $\left| J_{g^{-1}}(y) \right|$ is the determinant of the Jacobian matrix
+
+$$ \label{eq:jac}
+J_{g^{-1}}(y) = \left[ \begin{matrix}
+\frac{\mathrm{d}x_1}{\mathrm{d}y_1} & \ldots & \frac{\mathrm{d}x_1}{\mathrm{d}y_n} \\
+\vdots & \ddots & \vdots \\
+\frac{\mathrm{d}x_n}{\mathrm{d}y_1} & \ldots & \frac{\mathrm{d}x_n}{\mathrm{d}y_n}
+\end{matrix} \right]
+$$
+
+and $\mathcal{Y}$ is the set of possible outcomes of $Y$:
+
+$$ \label{eq:Y-range}
+\mathcal{Y} = \left\lbrace y = g(x): x \in \mathcal{X} \right\rbrace \; .
+$$
+
+In the present case, we have $Y = g(X) = X^\mathrm{T}$ and $X = g^{-1}(Y) = Y^\mathrm{T}$ and all $Y \in \mathcal{Y} = \mathbb{R}^{p \times n}$. For the vectorized matrices $X$ and $Y$, the Jacobian matrix is
+
+$$ \label{eq:XY-jac}
+J_{g^{-1}}(Y) = \left[ \begin{matrix}
+\frac{\mathrm{d}x_{11}}{\mathrm{d}y_{11}} & \frac{\mathrm{d}x_{11}}{\mathrm{d}y_{21}} & \ldots & \frac{\mathrm{d}x_{11}}{\mathrm{d}y_{pn}} \\
+\frac{\mathrm{d}x_{21}}{\mathrm{d}y_{11}} & \frac{\mathrm{d}x_{21}}{\mathrm{d}y_{21}} & \ldots & \frac{\mathrm{d}x_{21}}{\mathrm{d}y_{pn}} \\
+\vdots                                    & \vdots                                    & \ddots & \vdots                                    \\
+\frac{\mathrm{d}x_{np}}{\mathrm{d}y_{11}} & \frac{\mathrm{d}x_{np}}{\mathrm{d}y_{21}} & \ldots & \frac{\mathrm{d}x_{np}}{\mathrm{d}y_{pn}}
+\end{matrix} \right] \in \mathbb{R}^{np} \; .
+$$
+
+Because by transposition, $y_{ji} = x_{ij}$, we have
+
+$$ \label{eq:dxij-dyji}
+\frac{\mathrm{d}x_{ij}}{\mathrm{d}y_{kl}} = \left\{
+\begin{array}{rl}
+1 \; , & \text{if} \; k = j \; \text{and} \; l = i \\
+0 \; , & \text{otherwise} \; .
+\end{array}
+\right.
+$$
+
+Thus, $J_{g^{-1}}(Y)$ is row-equivalent to $I_{np}$ and $\left| J_{g^{-1}}(Y) \right| = \left| I_{np} \right| = 1$. Therefore, we have:
+
+$$
+\begin{split}
+   f_Y(Y)
+&= f_X(g^{-1}(Y)) \, \left| J_{g^{-1}}(Y) \right| \\
+&= \mathcal{MN}(Y^\mathrm{T}; M, U, V) \, \left| I_{np} \right| \\
+&= \mathcal{MN}(Y^\mathrm{T}; M, U, V) \; .
+\end{split}
+$$
+
+The [probability density function of the matrix-normal distribution](/P/matn-pdf) is:
 
 $$ \label{eq:matn-pdf-X}
 f(X) = \mathcal{MN}(X; M, U, V) = \frac{1}{\sqrt{(2\pi)^{np} |V|^n |U|^p}} \cdot \exp\left[-\frac{1}{2} \mathrm{tr}\left( V^{-1} (X-M)^\mathrm{T} \, U^{-1} (X-M) \right) \right] \; .
 $$
 
-Define $Y = X^\mathrm{T}$. Then, $X = Y^\mathrm{T}$ and we can substitute:
+Thus, substituting $X = Y^\mathrm{T}$, we get:
 
 $$ \label{eq:matn-pdf-Y-s1}
-f(Y) = \frac{1}{\sqrt{(2\pi)^{np} |V|^n |U|^p}} \cdot \exp\left[-\frac{1}{2} \mathrm{tr}\left( V^{-1} (Y^\mathrm{T}-M)^\mathrm{T} \, U^{-1} (Y^\mathrm{T}-M) \right) \right] \; .
+f(Y) = \frac{1}{\sqrt{(2\pi)^{np} |U|^p |V|^n}} \cdot \exp\left[-\frac{1}{2} \mathrm{tr}\left( V^{-1} (Y^\mathrm{T}-M)^\mathrm{T} \, U^{-1} (Y^\mathrm{T}-M) \right) \right] \; .
 $$
 
 Using $(A+B)^\mathrm{T} = (A^\mathrm{T} + B^\mathrm{T})$, we have:
 
 $$ \label{eq:matn-pdf-Y-s2}
-f(Y) = \frac{1}{\sqrt{(2\pi)^{np} |V|^n |U|^p}} \cdot \exp\left[-\frac{1}{2} \mathrm{tr}\left( V^{-1} (Y-M^\mathrm{T}) \, U^{-1} (Y-M^\mathrm{T})^\mathrm{T} \right) \right] \; .
+f(Y) = \frac{1}{\sqrt{(2\pi)^{np} |U|^p |V|^n}} \cdot \exp\left[-\frac{1}{2} \mathrm{tr}\left( V^{-1} (Y-M^\mathrm{T}) \, U^{-1} (Y-M^\mathrm{T})^\mathrm{T} \right) \right] \; .
 $$
 
 Using $\mathrm{tr}(ABC) = \mathrm{tr}(CAB)$, we obtain
 
 $$ \label{eq:matn-pdf-Y-s3}
-f(Y) = \frac{1}{\sqrt{(2\pi)^{np} |V|^n |U|^p}} \cdot \exp\left[-\frac{1}{2} \mathrm{tr}\left( U^{-1} (Y-M^\mathrm{T})^\mathrm{T} \, V^{-1} (Y-M^\mathrm{T}) \right) \right]
+f(Y) = \frac{1}{\sqrt{(2\pi)^{np} |U|^p |V|^n}} \cdot \exp\left[-\frac{1}{2} \mathrm{tr}\left( U^{-1} (Y-M^\mathrm{T})^\mathrm{T} \, V^{-1} (Y-M^\mathrm{T}) \right) \right]
 $$
 
-which is the [probability density function of a matrix-normal distribution](/P/matn-pdf) with mean $M^T$, covariance across rows $V$ and covariance across columns $U$.
+which is the [probability density function of a matrix-normal distribution](/P/matn-pdf) with mean $M^T$, covariance across rows $V$ and covariance across columns $U$:
+
+$$ \label{eq:matn-pdf-Y-s4}
+f(Y) = \mathcal{MN}(Y; M^\mathrm{T}, V, U) \; .
+$$