<s>[INST] 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?
 [/INST] Let $P_{ij}$ be a matrix that switches the $i$th and $j$th rows. Then $P^T_{ij}$ switches the $i$th and $j$th columns. Thus,  $M=P^T_{24}P_{13}N$. If you take $P=P^T_{24}P_{13}$, then $PNP^{-1}$= $P^T_{24}P_{13}N (P^T_{24}P_{13})^{-1}$=$M(P^T_{24}P_{13})^{-1}$ = $MP_{13}^{-1}(P^T_{24})^{-1}$. Switching rows and columns is its own inverse, so this is $MP_{13}P^T_{24}$. If the left-action of a matrix is to switch rows, then the right-action of that matrix is to switch columns. So $MP_{13}P^T_{24}$ means switch the 1st and 3rd columns of $M$, then switch the 2nd and 4th rows. But that doesn't affect $M$. So $PNP^{-1}$=$M$.
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