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 pa rametric representations of the boundary generating curve of the c-numerica l range of a matrix. Applying this result, we generalize the result of Ande rson to the c-numerical range. Furthermore, we give a description of the bo undary generating curve of the c-numerical range of certain types of nilpot ent Toeplitz matrices, A sufficient condition for the boundary generating c urve to be rational is obtained. Finally we explicitly compute the boundary generating curves of the numerical ranges for several concrete matrices an d classify the rationality of the curves. (C) 1999 Elsevier Science Inc. Al l rights reserved.