The q-numerical range of a reducible matrix via a normal operator

Let A be an n x n complex matrix and 0 <= q <= 1. The q-numerical range of A is the set denoted and defined by W-q(A) = {x*Ay : x, y is an element of C-n, vertical bar x vertical bar = vertical bar y vertical bar = 1, x*y = q}. We show that the q-numerical range of a reducible 3 x 3 matrix is determined by the q-numerical range of the normal operator (Tf) (z) = zf (z), f is an element of L-2 (Delta, dx dy) for some compact convex set Delta. The result provides a performable algorithm to compute the boundary of the q-numerical range of a reducible 3 x 3 matrix. An example is also given to illustrate the detail of computations of the boundary of the range. (c) 2006 Elsevier Inc. All rights reserved.