## 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_{1}T_{1}+⋯+x_{m}T_{m} on matrix tuples as L(X_{1},…,X_{m})=I⊗T_{0}+X_{1}⊗T_{1}+⋯+X_{m}⊗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×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.

Original language | English |
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Article number | 108888 |

Journal | Advances in Mathematics |

Volume | 415 |

DOIs | |

State | Published - 15 Feb 2023 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

- General Mathematics