Ranks of linear matrix pencils separate simultaneous similarity orbits

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 = T-0 + x(1)T(1) + middot middot middot + x(m)T(m) on matrix tuples as L(X-1, ..., X-m) = I circle times T-0 + X-1 circle times T-1 + middot middot middot + X-m circle times T-m. 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 x n matrices are simultaneously similar if and only if rk L(A) = rk L(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. (c) 2023 Elsevier Inc. All rights reserved.