(m \geq n\)</span>, defined by</p>
<p><span class="math display">\[
\frac{\partial}{\partial z_{m,n}}
\left( z \, z^{\top} \right)_{m,n}
\ = \
\frac{\partial}{\partial z_{m,n}}
\left( \sum_{k=1}^K z_{m,k} \, z_{n,k} \right)
\ = \
\left\{
\begin{array}{cl}
2 \, z_{n,n} &amp; \mbox{if } m = n \mbox{ and }
\\
z_{n,n} &amp; \mbox{if } m &gt; n.
\end{array}
\right.
\]</span></p>
<p>The absolute Jacobian determinant of the second transform is thus</p>
<p><span class="math display">\[
2^{K} \ \prod_{m = 1}^{K} \ \