Joint similarity of matrix tuples.

Abstract: This talk will discuss a classical problem in matrix theory: when are two tuples of matrices similar? We solve the two-sided version of the 2003 conjecture of Hadwin and Larson, itself an updated version of a 1985 conjecture of Curto and Herrero. Consider evaluations of linear pencils L = T_0 + x_1 T_1 + … + x_m T_m on matrix tuples using Kronecker’s tensor product by L(X_1,…,X_m) := I ⊗ T_0 + X_1 ⊗ T_1 + … + X_m ⊗ T_m. We will show that ranks of linear pencils constitute a collection of separating invariants for simultaneous similarity of matrix tuples. That is, m-tuples A and B of n × n matrices are simultaneously similar if and only if the ranks of L(A) and L(B) are equal for all linear matrix pencils L of size mn. Variants of this property exist for symplectic, orthogonal, unitary similarity, and for the left-right action of general linear groups. Finally, if time permits, a polynomial time algorithm for orbit equivalence of matrix tuples under the left-right action of special linear groups will be presented. The talk is based on joint work with Harm Derksen, Visu Makam and Jurij Volčič.