On some reciprocal matrices with elliptical components of their Kippenhahn curves

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.

Keywords

• numerical range
• kippenhahn curve
• reciprocal matrix
• primary 15a60. secondary 14h50
• 47b36