How to show that the following 2 matrices are conjugate?
\begin{bmatrix}
z & 0 \\
0
& z^{-1}
\end{bmatrix}
And
\begin{bmatrix}
z^{-1}& 0 \\
0 & z
\end{bmatrix}
I know the matrix that make them conjugate is this one:
\begin{bmatrix}
0 & 1 \\
-1 & 0
\end{bmatrix}
but I do not know the calculations that lead us to this matrix, could anyone show me the calculations please?
Best Answer
Actually, these two matrices are always conjugate: $$ \begin{bmatrix} u & 0 \\ 0 & v \end{bmatrix}, \qquad \begin{bmatrix} v & 0 \\ 0 & u \end{bmatrix} $$ and any invertible matrix of the form below conjugates them: $$ \begin{bmatrix} 0 & b \\ c & 0 \end{bmatrix} $$ Indeed, $$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} u & 0 \\ 0 & v \end{bmatrix} - \begin{bmatrix} v & 0 \\ 0 & u \end{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} a (u - v) & 0 \\ 0 & d (u - v) \end{bmatrix} $$ which gives $a=0, d=0$ if $u\ne v$.
Recall that two matrices $A$ and $B$ are conjugate iff there is an invertible matrix $P$ such that $PAP^{-1}=B$, or equivalently $PA-BP=0$.