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.
@article{bwmeta1.element.doi-10_2478_s11533-012-0150-3,
author = {Aikaterini Aretaki and Ioannis Maroulas},
title = {The higher rank numerical range of nonnegative matrices},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {435-446},
zbl = {1269.15019},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0150-3}
}
Aikaterini Aretaki; Ioannis Maroulas. The higher rank numerical range of nonnegative matrices. Open Mathematics, Tome 11 (2013) pp. 435-446. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0150-3/
[1] Aretaki Aik., Higher Rank Numerical Ranges of Nonnegative Matrices and Matrix Polynomials, PhD thesis, National Technical University of Athens, 2011 | Zbl 1326.15029
[2] Aretaki Aik., Maroulas J., The higher rank numerical range of matrix polynomials, preprint available at http://arxiv.org/abs/1104.1328 | Zbl 1326.15029
[3] Aretaki Aik., Maroulas J., The K-rank numerical radii, Ann. Funct. Anal., 2012, 3(1), 100–108 [Crossref] | Zbl 1273.47014
[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] | Zbl 1120.81011
[5] Gau H.-L., Li C.-K., Poon Y.-T., Sze N.-S., Higher rank numerical ranges of normal matrices, SIAM J. Matrix Anal. Appl., 2011, 32(1), 23–43 http://dx.doi.org/10.1137/09076430X[Crossref]
[6] Horn R.A., Johnson C.R., Matrix Analysis, Cambridge University Press, Cambridge, 1985 | Zbl 0576.15001
[7] Horn R.A., Johnson C.R., Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991 http://dx.doi.org/10.1017/CBO9780511840371[Crossref] | Zbl 0729.15001
[8] Issos J.N., The Field of Values of Non-Negative Irreducible Matrices, PhD thesis, Auburn University, 1966
[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] 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] | Zbl 1152.15027
[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] | Zbl 1003.15027
[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] | Zbl 1002.15028
[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] 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] | Zbl 0951.15022
[15] Woerdeman H.J., The higher rank numerical range is convex, Linear Multilinear Algebra, 2007, 56(1–2), 65–67 [WoS] | Zbl 1137.15018