The talk addresses the conjecture of Hadwin and Larson on joint similarity of matrix tuples, which arose in multivariate operator theory.

The main result states that the ranks of linear matrix pencils constitute a collection of separating invariants for joint similarity of matrix tuples, which affirmatively answers the two-sided version of the said conjecture. That is, m-tuples X and Y of n×n matrices are simultaneously similar if and only if rk L(X) = rk L(Y) for all linear matrix pencils L of size mn. Similar results hold for certain other group actions on matrix tuples. On the other hand, a pair of matrix tuples X and Y is given such that rk L(X) <= rk L(Y) for all L, but X does not lie in the closure of the joint similarity orbit of Y; this constitutes a counter-example to the general Hadwin-Larson conjecture.

The talk is based on joint work with Harm Derksen, Igor Klep and Visu Makam.