$ P = \left( \begin{array}{cccc}
\frac{n}{r} & 0 & 0 & 0 \\
0 & \frac{n}{t} & 0 & 0 \\
0 & 0 & -\frac{f+n}{f-n} & -\frac{2fn}{f-n} \\
0 & 0 & -1 & 0 \\
\end{array} \right) $

$ v $$ v' $
$ \left( \begin{array}{cccc}
\frac{n}{r} & 0 & 0 & 0 \\
0 & \frac{n}{t} & 0 & 0 \\
0 & 0 & -\frac{f+n}{f-n} & -\frac{2fn}{f-n} \\
0 & 0 & -1 & 0 \\
\end{array} \right) \left( \begin{array}{c}
x \\
y \\
z \\
w \\
\end{array} \right) = \left( \begin{array}{c}
x' \\
y' \\
z' \\
w' \\
\end{array} \right) $


\begin{flalign*}
x' &= x\frac{n}{r} \\
y' &= y\frac{n}{t} \\
z' &= -z\frac{f+n}{f-n} - w\frac{2fn}{f-n} \\
w' &= -z \\
\end{flalign*}

$ v $
\begin{flalign*}
x &= x'\frac{r}{n} \\
y &= y'\frac{t}{n} \\
z &= -w' \\
w &= -z'\frac{f-n}{2fn} + w'\frac{f+n}{2fn} \\
\end{flalign*}


$ P^{-1} = \left( \begin{array}{cccc}
\frac{r}{n} & 0 & 0 & 0 \\
0 & \frac{t}{n} & 0 & 0 \\
0 & 0 & 0 & -1 \\
0 & 0 & -\frac{f-n}{2fn} & \frac{f+n}{2fn} \\
\end{array} \right) = \left( \begin{array}{cccc}
\frac{1}{P_{c0,r0}} & 0 & 0 & 0 \\
0 & \frac{1}{P_{c1,r1}} & 0 & 0 \\
0 & 0 & 0 & -1 \\
0 & 0 & \frac{1}{P_{c3,r2}} & \frac{P_{c2,r2}}{P_{c3,r2}} \\
\end{array} \right) $