Ranks of linear matrix pencils separate simultaneous similarity orbits

Harm Derksen; Igor Klep; Visu Makam; Jurij Volčič

doi:10.1016/j.aim.2023.108888

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arXiv · 2109.09418

Ranks of linear matrix pencils separate simultaneous similarity orbits

Harm Derksen·Igor Klep·Visu Makam·Jurij Volčič

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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.

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Harm Derksen, Igor Klep, Visu Makam, Jurij Volčič. 2021-09-20. Ranks of linear matrix pencils separate simultaneous similarity orbits. https://doi.org/10.1016/j.aim.2023.108888

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