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Reduction of the \(c\)-numerical range to the classical numerical range. (English) Zbl 1210.15023
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 A. S. Lewis, P. A. Parrilo and 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 \(W_c(A)= W(B)\).

MSC:
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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