scholarly journals On some reciprocal matrices with elliptical components of their Kippenhahn curves

2021 ◽  
Vol 10 (1) ◽  
pp. 117-130
Author(s):  
Muyan Jiang ◽  
Ilya M. Spitkovsky
Keyword(s):  
Numerical Range ◽  
Main Diagonal ◽  
Higher Rank

Abstract By definition, reciprocal matrices are tridiagonal n-by-n matrices A with constant main diagonal and such that ai , i +1 ai +1, i = 1 for i = 1, . . ., n − 1. We establish some properties of the numerical range generating curves C(A) (also called Kippenhahn curves) of such matrices, in particular concerning the location of their elliptical components. For n ≤ 6, in particular, we describe completely the cases when C(A) consist entirely of ellipses. As a corollary, we also provide a complete description of higher rank numerical ranges when these criteria are met.

scholarly journals Central force and the higher rank numerical range

2012 ◽  
Vol 389 (1) ◽  
pp. 531-540 ◽  
Author(s):  
Mao-Ting Chien ◽  
Hiroshi Nakazato
Keyword(s):  
Numerical Range ◽  
Central Force ◽  
Higher Rank

Elliptical higher rank numerical range of some Toeplitz matrices

2018 ◽  
Vol 549 ◽  
pp. 256-275 ◽  
Author(s):  
Maria Adam ◽  
Aikaterini Aretaki ◽  
Ilya M. Spitkovsky
Keyword(s):  
Numerical Range ◽  
Toeplitz Matrices ◽  
Higher Rank

scholarly journals The higher rank numerical range of nonnegative matrices

2013 ◽  
Vol 11 (3) ◽  
Author(s):  
Aikaterini Aretaki ◽  
Ioannis Maroulas
Keyword(s):  
Complex Plane ◽  
Numerical Range ◽  
Nonnegative Matrix ◽  
Maximum Modulus ◽  
Companion Matrix ◽  
Nonnegative Matrices ◽  
Higher Rank

AbstractIn 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.

scholarly journals Multiaccess quantum communication and product higher rank numerical range

2013 ◽  
Vol 13 (7&8) ◽  
pp. 541-566
Author(s):  
Maciej Demianowicz ◽  
Pawel Horodecki ◽  
Karol Zyczkowski
Keyword(s):  
Multiple Access ◽  
Quantum Communication ◽  
Numerical Range ◽  
Analytical Techniques ◽  
Noise Model ◽  
Error Correction Codes ◽  
Multiple Access Channels ◽  
Exact Shape ◽  
Higher Rank ◽  
Quantum Error

In the present paper we initiate the study of the product higher rank numerical range. The latter, being a variant of the higher rank numerical range [M.--D. Choi {\it et al.}, Rep. Math. Phys. {\bf 58}, 77 (2006); Lin. Alg. Appl. {\bf 418}, 828 (2006)], is a natural tool for studying a construction of quantum error correction codes for multiple access channels. We review properties of this set and relate it to other numerical ranges, which were recently introduced in the literature. Further, the concept is applied to the construction of codes for bi--unitary two--access channels with a hermitian noise model. Analytical techniques for both outerbounding the product higher rank numerical range and determining its exact shape are developed for this case. Finally, the reverse problem of constructing a noise model for a given product range is considered.

The higher rank numerical range is convex

2008 ◽  
Vol 56 (1-2) ◽  
pp. 65-67 ◽  
Author(s):  
Hugo J. Woerdeman
Keyword(s):  
Numerical Range ◽  
Higher Rank

scholarly journals Condition for the higher rank numerical range to be non-empty

2009 ◽  
Vol 57 (4) ◽  
pp. 365-368 ◽  
Author(s):  
Chi-Kwong Li ◽  
Yiu-Tung Poon ◽  
Nung-Sing Sze
Keyword(s):  
Numerical Range ◽  
Higher Rank

Pauli group: Classification and joint higher rank numerical range

2018 ◽  
Vol 549 ◽  
pp. 136-152
Author(s):  
Hamid Reza Afshin ◽  
Mohammad Ali Mehrjoofard ◽  
Akbar Zare Chavoshi
Keyword(s):  
Numerical Range ◽  
Group Classification ◽  
Higher Rank

scholarly journals Line segments on the boundary of the numerical ranges of some tridiagonal matrices

2015 ◽  
Vol 30 ◽  
pp. 693-703 ◽  
Author(s):  
Ilya Spitkovsky ◽  
Claire Thomas
Keyword(s):  
Numerical Range ◽  
Main Diagonal ◽  
Geometric Progression ◽  
Diagonal Form ◽  
Line Segments ◽  
Tridiagonal Matrices

Tridiagonal matrices are considered for which the main diagonal consists of zeroes, the sup-diagonal of all ones, and the entries on the sub-diagonal form a geometric progression. The criterion for the numerical range of such matrices to have line segments on its boundary is established, and the number and orientation of these segments is described.

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