Generalized eigenvalue problems for meet and join matrices on semilattices

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Generalized eigenvalue problems for meet and join matrices on semilattices. / Ilmonen, Pauliina; Kaarnioja, Vesa.

In: Linear Algebra and Its Applications, Vol. 536, 01.01.2018, p. 250-273.

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@article{849523dfe2814912b0abed05fd4d5d8e,
title = "Generalized eigenvalue problems for meet and join matrices on semilattices",
abstract = "We study generalized eigenvalue problems for meet and join matrices with respect to incidence functions on semilattices. We provide new bounds for generalized eigenvalues of meet matrices with respect to join matrices under very general assumptions. The applied methodology is flexible, and it is shown in the case of GCD and LCM matrices that even sharper bounds can be obtained by applying the known properties of the divisor lattice. These results can also be easily modified for the dual problem of eigenvalues of join matrices with respect to meet matrices, which we briefly consider as well. We investigate the effectiveness of the obtained bounds for select examples involving number-theoretical lattices.",
keywords = "GCD matrix, Generalized eigenvalue, Meet matrix, Meet semilattice, M{\"o}bius function",
author = "Pauliina Ilmonen and Vesa Kaarnioja",
year = "2018",
month = "1",
day = "1",
doi = "10.1016/j.laa.2017.09.023",
language = "English",
volume = "536",
pages = "250--273",
journal = "Linear Algebra and Its Applications",
issn = "0024-3795",

}

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TY - JOUR

T1 - Generalized eigenvalue problems for meet and join matrices on semilattices

AU - Ilmonen, Pauliina

AU - Kaarnioja, Vesa

PY - 2018/1/1

Y1 - 2018/1/1

N2 - We study generalized eigenvalue problems for meet and join matrices with respect to incidence functions on semilattices. We provide new bounds for generalized eigenvalues of meet matrices with respect to join matrices under very general assumptions. The applied methodology is flexible, and it is shown in the case of GCD and LCM matrices that even sharper bounds can be obtained by applying the known properties of the divisor lattice. These results can also be easily modified for the dual problem of eigenvalues of join matrices with respect to meet matrices, which we briefly consider as well. We investigate the effectiveness of the obtained bounds for select examples involving number-theoretical lattices.

AB - We study generalized eigenvalue problems for meet and join matrices with respect to incidence functions on semilattices. We provide new bounds for generalized eigenvalues of meet matrices with respect to join matrices under very general assumptions. The applied methodology is flexible, and it is shown in the case of GCD and LCM matrices that even sharper bounds can be obtained by applying the known properties of the divisor lattice. These results can also be easily modified for the dual problem of eigenvalues of join matrices with respect to meet matrices, which we briefly consider as well. We investigate the effectiveness of the obtained bounds for select examples involving number-theoretical lattices.

KW - GCD matrix

KW - Generalized eigenvalue

KW - Meet matrix

KW - Meet semilattice

KW - Möbius function

UR - http://www.scopus.com/inward/record.url?scp=85030108273&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2017.09.023

DO - 10.1016/j.laa.2017.09.023

M3 - Article

VL - 536

SP - 250

EP - 273

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -

ID: 15719957