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?

Let's consider $\varphi:\mathbb C\rightarrow M_2(\mathbb C)$, $\varphi(a+ib)=\pmatrix{a & b \\ -b & a}$ the standard embedding of $\mathbb C$ into the matrix ring. 
Consider $Z(J)=\{A\in M_2(\mathbb C)\ | \ JA=AJ\}$ the set of the matrix commuting with $J$.
Your question is equivalent to show that $Z(J) = \varphi(\mathbb C)$.
And this is true because:
\begin{gather}
A=\pmatrix{a &b \\ c & d}\in Z(J) \Longleftrightarrow \pmatrix{a &b \\ c & d}\pmatrix{0 & 1 \\ -1 & 0} = \pmatrix{0 &1 \\ -1 & 0}\pmatrix{a &b \\ c & d} \Longleftrightarrow\\
\begin{cases}
-b = c\\
a=d
\end{cases}\Longleftrightarrow A=\pmatrix{a &b \\ -b & a}\in \varphi(\mathbb C)
\end{gather}
