The q-numerical range of 3 by 3 tridiagonal matrices

Abstract

For 0 ≤ q ≤ 1, we examine the q-numerical ranges of 3× 3 tridiagonal matrices A(b) that interpolate between the circular range W0(A(b)) and the elliptical range W1(A(b)) as q varies from 0 to 1. We show that for q ≤ (1 − b)2/(2(1 + b2)), Wq(A(b)) is a circular disc centered at the origin with radius (1 + b2)1/2, but W4/5(A(2)) is not even an elliptical disc.