On some reciprocal matrices with elliptical components of their Kippenhahn curves

Muyan Jiang; Ilya M. Spitkovsky

doi:10.1515/spma-2021-0151

On some reciprocal matrices with elliptical components of their Kippenhahn curves

Muyan Jiang, Ilya M. Spitkovsky

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

By definition, reciprocal matrices are tridiagonal n-by-n matrices A with constant main diagonal and such that ai,i+1ai+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.

Original language	English (US)
Pages (from-to)	117-130
Number of pages	14
Journal	Special Matrices
Volume	10
Issue number	1
DOIs	https://doi.org/10.1515/spma-2021-0151
State	Published - Jan 1 2022

Keywords

Kippenhahn curve
Numerical range
Reciprocal matrix

ASJC Scopus subject areas

Algebra and Number Theory
Geometry and Topology

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10.1515/spma-2021-0151

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abstract = "By definition, reciprocal matrices are tridiagonal n-by-n matrices A with constant main diagonal and such that ai,i+1ai+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. ",

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N2 - By definition, reciprocal matrices are tridiagonal n-by-n matrices A with constant main diagonal and such that ai,i+1ai+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.

AB - By definition, reciprocal matrices are tridiagonal n-by-n matrices A with constant main diagonal and such that ai,i+1ai+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.

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On some reciprocal matrices with elliptical components of their Kippenhahn curves

Abstract

Keywords

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