The q-numerical range of 3 by 3 tridiagonal matrices

Abstract

Abstract. For 0 ≤ q ≤ 1, we examine the q-numerical ranges of 3 × 3 tridiagonal matrices A(b) that interpolate between the circular range W 0 (A(b)) and the elliptical range W 1 (A(b)) as q varies from 0 to 1. We show that for q ≤ (1 − b) 2 /(2(1 + b 2 )), Wq(A(b)) is a circular disc centered at the origin with radius (1 + b 2 ) 1/2 , but W 4/5 (A(2)) is not even an elliptical disc.