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

Let A be an n × n complex matrix and 0 q 1. The q-numerical range of A is the set denoted and defined byWq(A)={x*Ay:x,y Cn,x=y=1,x*y=q}.We show that the q-numerical range of a reducible 3 × 3 matrix is determined by the q-numerical range of the normal operator for some compact convex set Δ. The result provides a performable algorithm to compute the boundary of the q-numerical range of a reducible 3 × 3 matrix. An example is also given to illustrate the detail of computations of the boundary of the range.