We initiate the study of the complexity of arithmetic circuits with division gates over non-commuting variables. Such circuits and formulae compute non-commutative rational functions, which, despite their name, can no longer be expressed as ratios of polynomials. We prove some lower and upper bounds, completeness and simulation results, as follows.

If $X$ is an $n\times n$ matrix consisting of $n^2$ distinct mutually non-commuting variables, we show that: $X^{-1}$ can be computed by a circuit of polynomial size. Every formula computing some entry of $X^{-1}$ must have size at least $2^{\Omega(n)}$. We also show that matrix inverse is complete in the following sense: Assume that a non-commutative rational function $f$ can be computed by a formula of size $s$. Then there exists an invertible $2s\times 2s$-matrix $A$ whose entries are variables or field elements such that $f$ is an entry of $A^{-1}$. If $f$ is a non-commutative polynomial computed by a formula without inverse gates then $A$ can be taken as an upper triangular matrix with field elements on the diagonal. We show how divisions can be eliminated from non-commuta\-tive circuits and formulae which compute polynomials, and we address the non-commutative version of the “rational function identity testing” problem. As it happens, the complexity of both of these procedures depends on a single open problem in invariant theory