Alexandru Nica, Dimitri Shlyakhtenko, and Roland Speicher
\newcommand{\A}{ {\cal A} } \newcommand{\C}{ {\bf C} }
We introduce the concept of ``R-cyclic family'' of matrices with entries in a non-commutative probability space; the definition consists in asking that only the ``cyclic'' non-crossing cumulants of the entries of the matrices are allowed to be non-zero.
Let $A_{1}, \ldots , A_{s}$ be an R-cyclic family of $d \times d$ matrices over a non-commutative probability space $\ncps$. We prove a convolution-type formula for the explicit computation of the joint distribution of $A_{1}, \ldots , A_{s}$ (considered in $M_{d} ( \A )$ with the natural state), in terms of the joint distribution (considered in the original space $\ncps$) of the entries of the $s$ matrices. Several important situations of families of matrices with tractable joint distributions arise by application of this formula.
Moreover, let $A_{1}, \ldots , A_{s}$ be a family of $d \times d$ matrices over a non-commutative probability space $\ncps$, let $\D \subset M_{d} ( \A )$ denote the algebra of scalar diagonal matrices, and let ${\cal C}$ be the subalgebra of $M_{d} ( \A )$ generated by $\{ A_{1}, \ldots , A_{s} \} \cup \D$. We prove that the R-cyclicity of $A_{1}, \ldots , A_{s}$ is equivalent to a property of ${\cal C}$ -- namely that ${\cal C}$ is free from $M_{d} ( \C )$, with amalgamation over $\D$.