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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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##### References:
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