Spectral properties of positively homogeneous operators induced by higher order tensors.

Yisheng Song; Liqun Qi

doi:10.1137/130909135

Spectral properties of positively homogeneous operators induced by higher order tensors.

Yisheng Song, Liqun Qi

Research output: Journal article publication › Journal article › Academic research › peer-review

60 Citations (Scopus)

Abstract

The Fredholm alternative type results are proved for eigenvalues (E-eigenvalues, H-eigenvalues, Z-eigenvalues) of a higher order tensor A. For the positively homogeneous operators FA and TA induced by a higher order tensor A, we show some relationship between the Gelfand formula and the spectral radius, and present the upper bound of their spectral radii. Furthermore, for a nonnegative tensor A, we obtain the practical relevance for the spectral radius of the operators FA and TA as well as the operator norms of FA and TA.

Original language	English
Pages (from-to)	1581-1595
Number of pages	15
Journal	SIAM Journal on Matrix Analysis and Applications
Volume	34
Issue number	4
DOIs	https://doi.org/10.1137/130909135
Publication status	Published - 1 Dec 2013

Keywords

Eigenvalue-eigenvector
Fredholm alternative
Gelfand formula
Higher order tensor
Positively homogeneous

ASJC Scopus subject areas

Analysis

More information

10.1137/130909135

Cite this

@article{b083b7ceecea4ee29528e02ef6ba9382,

title = "Spectral properties of positively homogeneous operators induced by higher order tensors.",

abstract = "The Fredholm alternative type results are proved for eigenvalues (E-eigenvalues, H-eigenvalues, Z-eigenvalues) of a higher order tensor A. For the positively homogeneous operators FA and TA induced by a higher order tensor A, we show some relationship between the Gelfand formula and the spectral radius, and present the upper bound of their spectral radii. Furthermore, for a nonnegative tensor A, we obtain the practical relevance for the spectral radius of the operators FA and TA as well as the operator norms of FA and TA.",

keywords = "Eigenvalue-eigenvector, Fredholm alternative, Gelfand formula, Higher order tensor, Positively homogeneous",

author = "Yisheng Song and Liqun Qi",

year = "2013",

month = dec,

day = "1",

doi = "10.1137/130909135",

language = "English",

volume = "34",

pages = "1581--1595",

journal = "SIAM Journal on Matrix Analysis and Applications",

issn = "0895-4798",

publisher = "Society for Industrial and Applied Mathematics Publications",

number = "4",

}

TY - JOUR

T1 - Spectral properties of positively homogeneous operators induced by higher order tensors.

AU - Song, Yisheng

AU - Qi, Liqun

PY - 2013/12/1

Y1 - 2013/12/1

N2 - The Fredholm alternative type results are proved for eigenvalues (E-eigenvalues, H-eigenvalues, Z-eigenvalues) of a higher order tensor A. For the positively homogeneous operators FA and TA induced by a higher order tensor A, we show some relationship between the Gelfand formula and the spectral radius, and present the upper bound of their spectral radii. Furthermore, for a nonnegative tensor A, we obtain the practical relevance for the spectral radius of the operators FA and TA as well as the operator norms of FA and TA.

AB - The Fredholm alternative type results are proved for eigenvalues (E-eigenvalues, H-eigenvalues, Z-eigenvalues) of a higher order tensor A. For the positively homogeneous operators FA and TA induced by a higher order tensor A, we show some relationship between the Gelfand formula and the spectral radius, and present the upper bound of their spectral radii. Furthermore, for a nonnegative tensor A, we obtain the practical relevance for the spectral radius of the operators FA and TA as well as the operator norms of FA and TA.

KW - Eigenvalue-eigenvector

KW - Fredholm alternative

KW - Gelfand formula

KW - Higher order tensor

KW - Positively homogeneous

UR - http://www.scopus.com/inward/record.url?scp=84892395668&partnerID=8YFLogxK

U2 - 10.1137/130909135

DO - 10.1137/130909135

M3 - Journal article

SN - 0895-4798

VL - 34

SP - 1581

EP - 1595

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 4

ER -

Spectral properties of positively homogeneous operators induced by higher order tensors.

Abstract

Keywords

ASJC Scopus subject areas

More information

Other files and links

Fingerprint

Cite this