Orthogonality graphs of matrices over skew fields

The paper is devoted to studying the orthogonality graph of the matrix ring over a skew field. It is shown that for $n\geq3$ the orthogonality graph of the $n\times n$ matrix ring $M_n(\mathbb D)$ over a skew field $\mathbb D$ is connected and has diameter $4$ for an arbitrary skew field $\mathbb D$. If $n=2$, then the graph of the ring $M_n(\mathbb D)$ is a disjoint union of connected components of diameters $1$ and $2$. As a corollary, we obtain related results on the orthogonality graphs of simple Artinian rings.