The numerical range of certain 0,1-matrices †

Let A be a 0, 1-matrix with at most one 1 in each row and column. The authors prove that the numerical range of A is the convex hull of a polygon and a circular disk, both centered at the origin and contained in the unit disk. The proof uses a permutation similarity to reduce A to a direct sum of matrices whose numerical ranges can be determined. A computer program, developed by the authors, which plots the boundary of the numerical range of an arbitrary complex matrix is also discussed.