Reduction of the c-numerical range to the classical numerical range

Let A be an n x n complex matrix and c = (c(1), c(2), . . . , c(n)) a real n-tuple. The c-numerical range of A is defined as the set W(c)(A) = {Sigma(n)(j=1) c(j)x(j)*Ax(j) : {x(1), x(2), . . . , x(n)} an orthonormal basis for C(n)} when c = (1, 0, . . . , 0), W(c)(A) becomes the classical numerical range of A which is often defined as the set W(A) = {x*Ax : x is an element of C(n), x*x = 1}. We show that for any n x n complex matrix A and real n-tuple c, there exists a complex matrix B of size at most n ! such that W(c) (A) = W(B). Constructions of the matrix B for some matrices A and real n-tuple c are provided. (C) 2010 Elsevier Inc. All rights reserved.