Why matrices commuting with $\small\begin{bmatrix} 0&1\\-1&0\end{bmatrix}$ represent complex numbers? I am trying to understand which $2$ by $2$ real matrices represent complex numbers in following way.
Let $J=\begin{bmatrix} 0&1\\-1&0\end{bmatrix}$ and $A=\begin{bmatrix} a&b\\c&d\end{bmatrix}$ be any real matrix.
If $A$ represents a complex matrix (by standard embedding of complex field into matrix ring) then $A$ should commute with the matrix $J$, which image of complex number $i$.
Q. I want to understand why the matrices commuting with $J$ are precisely the matrices representing complex numbers?

$L : X \longmapsto Y$ $\mathbb{R}$-linear application between complex vectorial space is $\mathbb{C}$-linear iff commutes with $i$.
Infact $L$ it's $\mathbb{C}$-linear $\iff$ $iL = Li \iff J_{Y}L = LJ_{X}$
