Journal article
Ranks of linear matrix pencils separate simultaneous similarity orbits
Advances in mathematics (New York. 1965), v 415, 108888
15 Feb 2023
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=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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Details
- Title
- Ranks of linear matrix pencils separate simultaneous similarity orbits
- Creators
- Harm Derksen - Northeastern UniversityIgor Klep - Institute of Mathematics, Physics, and MechanicsVisu MakamJurij Volčič - Drexel University, Mathematics
- Publication Details
- Advances in mathematics (New York. 1965), v 415, 108888
- Publisher
- Elsevier
- Grant note
- DMS-1954709 / National Science Foundation (https://doi.org/10.13039/100000001) IIS-1837985; DMS-2001460 / National Science Foundation (https://doi.org/10.13039/100000001) CCF-1900460 / National Science Foundation (https://doi.org/10.13039/100000001) J1-2453; N1-0217; J1-3004; P1-0222 / Slovenian Research Agency (https://doi.org/10.13039/501100004329) University of Melbourne (https://doi.org/10.13039/501100001782) J1-3004 / Slovenian Research Agency (https://doi.org/10.13039/501100004329)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000933292000001
- Scopus ID
- 2-s2.0-85147816003
- Other Identifier
- 991021861180104721
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- Collaboration types
- Domestic collaboration
- International collaboration
- Web of Science research areas
- Mathematics