A Model Theoretic Perspective on Matrix Rings

In this paper natural necessary and sufficient conditions for quantifier elimination of matrix rings $M_n(K)$ in the language of rings expanded by two unary functions, naming the trace and transposition, are identified. This is used together with invariant theory to prove quantifier elimination when $K$ is an intersection of real closed fields. On the other hand, it is shown that finding a natural \textit{definable} expansion with quantifier elimination of the theory of $M_n({\mathbb C})$ is closely related to the infamous simultaneous conjugacy problem in matrix theory. Finally, for various natural structures describing dimension-free matrices it is shown that no such elimination results can hold by establishing undecidability results.