On Kippenhahn curves and higher-rank numerical ranges of some matrices

Bebiano, Natália; Providência, J. da; Spitkovsky, Ilya M.

doi:10.48550/arxiv.2104.07893

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Reciprocal 4-by-4 and 5-by-5 Matrices1

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2021

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“…To illustrate, consider A ∈ RM 5 with Observe also that under condition (3.17) A is what in [1] was called a 2-periodic reciprocal matrix. Such matrices of dimension n = 1 mod 4 (in particular, n = 5) have no elliptical components in C(A) [1, Theorem 5].…”

Section: Reciprocal 4-by-4 and 5-by-5 Matricesmentioning

confidence: 99%

On some reciprocal matrices with elliptical components of their Kippenhahn curves

Jiang

Spitkovsky

2021

Special Matrices

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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.

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Section: Reciprocal 4-by-4 and 5-by-5 Matricesmentioning

confidence: 99%