Ranks of linear matrix pencils separate simultaneous similarity orbits
Abstract
This paper solves the twosided 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 leftright action of general linear groups. Furthermore, a polynomial time algorithm for orbit equivalence of matrix tuples under the leftright action of special linear groups is deduced.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.09418
 Bibcode:
 2021arXiv210909418D
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  Functional Analysis;
 Mathematics  Representation Theory