Es ist

${\displaystyle {}M_{\mathfrak {u}}^{\mathfrak {v}}={\begin{pmatrix}1&0&-1\\4&1&1\\5&2&0\end{pmatrix}}\,.}$

Für die umgekehrte Übergangsmatrix müssen wir diese Matrix invertieren. Es ist

${\displaystyle {}{\begin{pmatrix}1&0&-1\\4&1&1\\5&2&0\end{pmatrix}}}$	${\displaystyle {}{\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}}$
${\displaystyle {}{\begin{pmatrix}1&0&-1\\0&1&5\\0&2&5\end{pmatrix}}}$	${\displaystyle {}{\begin{pmatrix}1&0&0\\-4&1&0\\-5&0&1\end{pmatrix}}}$
${\displaystyle {}{\begin{pmatrix}1&0&-1\\0&1&5\\0&0&-5\end{pmatrix}}}$	${\displaystyle {}{\begin{pmatrix}1&0&0\\-4&1&0\\3&-2&1\end{pmatrix}}}$
${\displaystyle {}{\begin{pmatrix}1&0&-1\\0&1&5\\0&0&1\end{pmatrix}}}$	${\displaystyle {}{\begin{pmatrix}1&0&0\\-4&1&0\\-{\frac {3}{5}}&{\frac {2}{5}}&-{\frac {1}{5}}\end{pmatrix}}}$
${\displaystyle {}{\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}}$	${\displaystyle {}{\begin{pmatrix}{\frac {2}{5}}&{\frac {2}{5}}&-{\frac {1}{5}}\\-1&-1&1\\-{\frac {3}{5}}&{\frac {2}{5}}&-{\frac {1}{5}}\end{pmatrix}}}$

Es ist also

${\displaystyle {}M_{\mathfrak {v}}^{\mathfrak {u}}={\begin{pmatrix}{\frac {2}{5}}&{\frac {2}{5}}&-{\frac {1}{5}}\\-1&-1&1\\-{\frac {3}{5}}&{\frac {2}{5}}&-{\frac {1}{5}}\end{pmatrix}}\,.}$