Matrix Equation of three matrices.

Question

Suppose I have square matrices $A$, $B$ and $C$ each of dimension $n \times n$ in $\mathbb{M}_n({\mathbb{C}})$. These are related by the following equations: \begin{equation} \begin{split} AB - BA &= iC\\ BC - CB &= iA\\ CA - AC &= iB\\ \end{split} \end{equation} where $i = \sqrt{\left(-1\right)}$. Is it possible to obtain non trivial solution of such a system? Will the solution be unique? And does the dimension play a role here?

Answer 1

There are several solutions. Clearly $A=B=C=0$ is always a solution. But, say, for $n=3$ we also have the matrices $A,B,C$ of the simple Lie algebra $\mathfrak{so}_3(\Bbb C)$, given by $$ A=\begin{pmatrix} 0 & i & 0\cr -i & 0 & 0\cr 0 & 0 & 0\end{pmatrix}, \; B=\begin{pmatrix} 0 & 0 & i\cr 0 & 0 & 0\cr -i & 0 & 0\end{pmatrix}, \; C=\begin{pmatrix} 0 & 0 & 0\cr 0 & 0 & -i\cr 0 & i & 0\end{pmatrix}. \; $$ They satisfy the three equations. This can be generalised.