Boundary generating curves of the c-numerical range

Let A be an n x n matrix and c = (c(1), c(2), ..., c(n)) be a real n-tuple. The c-numerical range of A is the set W-c(A) = {Sigma(j=1)(n), c(j)x(j)*Ax(j) : {x(1), x(2), ..., x(n)} is an orthonormal basis of C-n}. We obtain parametric representations of the boundary generating curve of the c-numerical range of a matrix. Applying this result, we generalize the result of Anderson to the c-numerical range. Furthermore, we give a description of the boundary generating curve of the c-numerical range of certain types of nilpotent Toeplitz matrices, A sufficient condition for the boundary generating curve to be rational is obtained. Finally we explicitly compute the boundary generating curves of the numerical ranges for several concrete matrices and classify the rationality of the curves. (C) 1999 Elsevier Science Inc. All rights reserved.