Ranks of linear matrix pencils separate simultaneous similarity orbits

Abstract

This paper solves the two-sided version and provides a counterexample to the general version of the 2003 conjecture by Hadwin and Larson. Consider evaluations of linear matrix pencils L=T0+x1T1+⋯+xmTm on matrix tuples as L(X1,…,Xm)=I⊗T0+X1⊗T1+⋯+Xm⊗Tm. It is shown that ranks of linear matrix 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 rkL(A)=rkL(B) for all linear matrix pencils L of size mn. Variants of this property are also established for symplectic, orthogonal, unitary similarity, and for the left-right action of general linear groups. Furthermore, a polynomial time algorithm for orbit equivalence of matrix tuples under the left-right action of special linear groups is deduced.