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 +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.

Invitation to convex algebraic geometry

“…The shape of the numerical range is well understood [16] for matrices of sizes n = 2 and n = 3. The shape remains a research topic in matrix theory [4,1,15] and quantum information theory [9] for matrices of sizes n ≥ 4. Among the basic key-theorems is the Toeplitz-Hausdorff theorem [26,10] from the years 1918/19, which asserts that the numerical range W (A) is a convex set, and Kippenhahn's assertion [17,Sec.…”

“…10] from 1951, which states that W (A) is the convex hull of a plane real algebraic curve C A . The curve C A has been called boundary generating curve [17,3], and was renamed to Kippenhahn curve [6,8,1,15] recently.…”