How do I find out that the following two matrices are similar?

Let $(e_1,e_2,e_3,e_3)$ be the standard basis of $\mathbb{R}^4$. You have:

• $N.e_1=0$;
• $N.e_2=e_1$;
• $N.e_3=0$;
• $N.e_4=0$.

You also have:

• $M.e_3=0$;
• $M.e_4=e_3$;
• $M.e_1=0$;
• $M.e_2=0$.

So, if you see $M$ as a linear map from $\mathbb{R}^4$ into itself, the matrix of $M$ with respect to the basis $(e_3,e_4,e_1,e_2)$ is the matrix $N$. Therefore, $N$ and $M$ are similar.

Or you can take$$P=\begin{bmatrix}0&0&1&0\\0&0&0&1\\1&0&0&0\\0&1&0&0\end{bmatrix},$$which is basically the same thing.

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: $$ M=\begin{bmatrix} 0 & I_2 \\ I_2 & 0 \end{bmatrix} N \begin{bmatrix} 0 & I_2 \\ I_2 & 0 \end{bmatrix} $$