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Ranks of linear matrix pencils separate simultaneous similarity orbits
Journal article   Open access   Peer reviewed

Ranks of linear matrix pencils separate simultaneous similarity orbits

Harm Derksen, Igor Klep, Visu Makam and Jurij Volčič
Advances in mathematics (New York. 1965), v 415, 108888
15 Feb 2023
DOI: https://doi.org/10.1016/j.aim.2023.108888
url
https://doi.org/10.1016/j.aim.2023.108888View
Accepted (AM)Maybe Open Access (Publisher Bronze),  Restricted

Abstract

Linear matrix pencil Module degeneration Orbit equivalence Rank-preserving map Simultaneous similarity
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.

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