Ranks of linear matrix pencils separate simultaneous similarity orbits

TL;DR

This paper demonstrates that ranks of linear matrix pencils serve as complete invariants for simultaneous similarity of matrix tuples, providing new theoretical insights and an efficient algorithm for orbit classification.

Contribution

It proves that ranks of linear matrix pencils uniquely determine simultaneous similarity classes and extends this to various group actions, also offering a polynomial-time algorithm for orbit equivalence.

Findings

Ranks of linear matrix pencils separate simultaneous similarity orbits.

Counterexample provided for the general version of Hadwin and Larson's conjecture.

Polynomial time algorithm for orbit equivalence under certain group actions.

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 $L=T_0+x_1T_1+\cdots+x_mT_m$ on matrix tuples as $L(X_1,\dots,X_m)=I\otimes T_0+X_1\otimes T_1+\cdots+X_m\otimes T_m$. 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 $n\times n$ 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 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.