# 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?

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

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 $$PNP^{-1}$$=$$M$$.