Reduction of the \(c\)-numerical range to the classical numerical range
For an \(n\)-by-\(n\) complex matrix \(A\) and a real \(n\)-tuple \(c=(c_1,\dots, c_n)\), the \(c\)-numerical range \(W_c(A)\) of \(A\) is, by definition, the subset \[ \Biggl\{\sum^n_{j=1} c_j x^*_j Ax_j: x_1,\dots, x_n\text{ form an orthonormal basis of }\mathbb{C}^n\Biggr\} \] of the complex plane. If \(c= (1,0,\dots,0)\), then \(W_c(A)\) is the classical numerical range \(W(A)\) of \(A\). The main result of this paper is that every \(W_c(A)\) is equal to \(W(B)\) for some symmetric matrix \(B\) of size at most \(n!\). This is proven using the confirmed Lax conjecture by \textit{A. S. Lewis}, \textit{P. A. Parrilo} and \textit{M. V. Ramana} [Proc. Am. Math. Soc. 133, No. 9, 2495--2499 (2005; Zbl 1073.90029)] or, more precisely, a special case of it, the Fiedler conjecture, that if \(p(x,y,z)\) is a homogeneous polynomial of degree \(n\) with the hyperbolicity property, namely, \(p(1,0,0)= 1\) and \(p(w_1- t,w_2,w_3)= 0\) has only real roots for any real \(w_1\), \(w_2\) and \(w+3\), then \(p(x,y,z)= \text{det}(xI_n+ yH_1+ zH_2)\) for some \(n\)-by-\(n\) Hermitian matrices \(H_1\) and \(H_2\). The connection between hyperbolic polynomials and numerical ranges is Kippenhahn's result that \(W(A)\) equals the convex hull of the curve \(u+iv\), where \(u\) and \(v\) are real such that \(1+ uy+ vz= 0\) is a tangent line of the curve \(\text{det}(I_n+ u\,\text{Re\,}A + v\,\text{Im\,}A)= 0\) (\(\text{Re\,}A= (A+ A^*)/2\) and \(\text{Im\,}A= (A- A^*)/(2i)\) are the real and imaginary parts of \(A\), respectively). For illustrations, the authors construct explicitly, for certain 3-by-3 and 4-by-4 matrices \(A\) and certain tuples \(c\), the matrix \(B\) for which \(W_c(A)= W(B)\).
scientific article; zbMATH DE number 5833969
scientific article