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}.Wq(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(Tf)(z)=zf(z),f∈L2(Δ,dxdy)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.