The higher rank numerical range of nonnegative matrices

Aikaterini Aretaki; Ioannis Maroulas

Open Mathematics (2013)

  • Volume: 11, Issue: 3, page 435-446
  • ISSN: 2391-5455

Abstract

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In this article the rank-k numerical range ∧k (A) of an entrywise nonnegative matrix A is investigated. Extending the notion of elements of maximum modulus in ∧k (A), we examine their location on the complex plane. Further, an application of this theory to ∧k (L(λ)) of a Perron polynomial L(λ) is elaborated via its companion matrix C L.

How to cite

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Aikaterini Aretaki, and Ioannis Maroulas. "The higher rank numerical range of nonnegative matrices." Open Mathematics 11.3 (2013): 435-446. <http://eudml.org/doc/268979>.

@article{AikateriniAretaki2013,
abstract = {In this article the rank-k numerical range ∧k (A) of an entrywise nonnegative matrix A is investigated. Extending the notion of elements of maximum modulus in ∧k (A), we examine their location on the complex plane. Further, an application of this theory to ∧k (L(λ)) of a Perron polynomial L(λ) is elaborated via its companion matrix C L.},
author = {Aikaterini Aretaki, Ioannis Maroulas},
journal = {Open Mathematics},
keywords = {Perron-Frobenius theory; Nonnegative matrix; Perron polynomial; Higher rank numerical range; Rank-k numerical radius; nonnegative matrix; higher rank numerical range; rank- numerical radius},
language = {eng},
number = {3},
pages = {435-446},
title = {The higher rank numerical range of nonnegative matrices},
url = {http://eudml.org/doc/268979},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Aikaterini Aretaki
AU - Ioannis Maroulas
TI - The higher rank numerical range of nonnegative matrices
JO - Open Mathematics
PY - 2013
VL - 11
IS - 3
SP - 435
EP - 446
AB - In this article the rank-k numerical range ∧k (A) of an entrywise nonnegative matrix A is investigated. Extending the notion of elements of maximum modulus in ∧k (A), we examine their location on the complex plane. Further, an application of this theory to ∧k (L(λ)) of a Perron polynomial L(λ) is elaborated via its companion matrix C L.
LA - eng
KW - Perron-Frobenius theory; Nonnegative matrix; Perron polynomial; Higher rank numerical range; Rank-k numerical radius; nonnegative matrix; higher rank numerical range; rank- numerical radius
UR - http://eudml.org/doc/268979
ER -

References

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  1. [1] Aretaki Aik., Higher Rank Numerical Ranges of Nonnegative Matrices and Matrix Polynomials, PhD thesis, National Technical University of Athens, 2011 Zbl1326.15029
  2. [2] Aretaki Aik., Maroulas J., The higher rank numerical range of matrix polynomials, preprint available at http://arxiv.org/abs/1104.1328 Zbl1326.15029
  3. [3] Aretaki Aik., Maroulas J., The K-rank numerical radii, Ann. Funct. Anal., 2012, 3(1), 100–108 [Crossref] Zbl1273.47014
  4. [4] Choi M.-D., Kribs D.W., Zyczkowski K., Quantum error correcting codes from the compression formalism, Rep. Math. Phys., 2006, 58(1), 77–91 http://dx.doi.org/10.1016/S0034-4877(06)80041-8[Crossref] Zbl1120.81011
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  7. [7] Horn R.A., Johnson C.R., Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991 http://dx.doi.org/10.1017/CBO9780511840371[Crossref] Zbl0729.15001
  8. [8] Issos J.N., The Field of Values of Non-Negative Irreducible Matrices, PhD thesis, Auburn University, 1966 
  9. [9] Li C.-K., Poon Y.-T., Sze N.-S., Condition for the higher rank numerical range to be non-empty, Linear Multilinear Algebra, 2009, 57(4), 365–368 http://dx.doi.org/10.1080/03081080701786384[Crossref][WoS] 
  10. [10] Li C.-K., Sze N.-S., Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations, Proc. Amer. Math. Soc., 2008, 136(9), 3013–3023 http://dx.doi.org/10.1090/S0002-9939-08-09536-1[Crossref][WoS] Zbl1152.15027
  11. [11] Li C.-K., Tam B.-S., Wu P.Y., The numerical range of a nonnegative matrix, Linear Algebra Appl., 2002, 350, 1–23 http://dx.doi.org/10.1016/S0024-3795(02)00291-4[Crossref] Zbl1003.15027
  12. [12] Maroulas J., Psarrakos P.J., Tsatsomeros M.J., Perron-Frobenius type results on the numerical range, Linear Algebra Appl., 2002, 348, 49–62 http://dx.doi.org/10.1016/S0024-3795(01)00574-2[Crossref] Zbl1002.15028
  13. [13] Psarrakos P.J., Tsatsomeros M.J., A primer of Perron-Frobenius theory for matrix polynomials, Linear Algebra Appl., 2004, 393, 333–351 http://dx.doi.org/10.1016/j.laa.2003.12.026[Crossref] 
  14. [14] Tam B.-S., Yang S., On matrices whose numerical ranges have circular or weak circular symmetry, Linear Algebra Appl., 1999, 302/303, 193–221 http://dx.doi.org/10.1016/S0024-3795(99)00174-3[Crossref] Zbl0951.15022
  15. [15] Woerdeman H.J., The higher rank numerical range is convex, Linear Multilinear Algebra, 2007, 56(1–2), 65–67 [WoS] Zbl1137.15018

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