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 $L=T_0+x_1T_1+\cdots+x_mT_m$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2021
- DOI:
- 10.48550/arXiv.2109.09418
- arXiv:
- arXiv:2109.09418
- Bibcode:
- 2021arXiv210909418D
- Keywords:
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- Mathematics - Rings and Algebras;
- Mathematics - Functional Analysis;
- Mathematics - Representation Theory
- E-Print:
- Adv. Math. 415 (2023) 108888, 20pp