How do I find out that the following two matrices are similar? How do I find out that the following two matrices are similar?
$N =
\begin{pmatrix} 
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 
\end{pmatrix}$
and $M=
\begin{pmatrix} 
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 
\end{pmatrix}$
I initially tried to think of the left multiplication of a matrix $P$ as a row operation and tried
$P=
\begin{pmatrix} 
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 
\end{pmatrix}$
such that $PN = \begin{pmatrix} 
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 
\end{pmatrix}$ but then $PNP^{-1} \neq M$.
My linear algebra is a bit rusty. Is there a more elaborate way to do this?

The two matrices are made of Jordan blocks; in $2\times2$ block format, they are
$$
N=\begin{bmatrix} J & 0 \\ 0 & 0 \end{bmatrix}
\qquad
M=\begin{bmatrix} 0 & 0 \\ 0 & J \end{bmatrix}
$$
You get similar matrices if you perform a row switch together with the corresponding column switch; in this case there is only one possible switch:
$$
M=\begin{bmatrix} 0 & I_2 \\ I_2 & 0 \end{bmatrix} N
\begin{bmatrix} 0 & I_2 \\ I_2 & 0 \end{bmatrix}
$$
