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

Let A   be an n×nn×n complex matrix and c=(c1,c2,…,cn)c=(c1,c2,…,cn) a real n-tuple. The c-numerical range of A is defined as the setWc(A)=∑j=1ncjxj∗Axj:{x1,x2,…,xn}isanorthonormalbasisforCn.When c=(1,0,…,0)c=(1,0,…,0), Wc(A)Wc(A) becomes the classical numerical range of A which is often defined as the setW(A)={x∗Ax:x∈Cn,x∗x=1}.W(A)={x∗Ax:x∈Cn,x∗x=1}.We show that for any n×nn×n complex matrix A and real n-tuple c, there exists a complex matrix B   of size at most n!n! such that Wc(A)=W(B).Wc(A)=W(B). Constructions of the matrix B for some matrices A and real n-tuple c are provided.