Boundary generating curves of the c-numerical range Original Research Article

Let A be an n×n matrix and c=(c1,c2,…,cn) be a real n-tuple. The c-numerical range of A is the set Wc(A)={∑j=1ncjxj*Axj:{x1,x2,…,xn} is an orthonormal basis of image. 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.