## 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

Zbl 1073.90029
Full Text:

### References:

 [1] Au-Yeung, Y. H.; Tsing, N. K., An extension of the Hausdorff-Toeplitz theorem on the numerical range, Proc. Amer. Math. Soc., 89, 215-218 (1983) · Zbl 0525.47002 [2] Chien, M. T.; Nakazato, H., Boundary generating curves of the $$c$$-numerical range, Linear Algebra Appl., 294, 67-84 (1999) · Zbl 0932.15017 [3] Chien, M. T.; Nakazato, H., The $$c$$-numerical range of tridiagonal matrices, Linear Algebra Appl., 335, 55-61 (2001) · Zbl 0982.15034 [4] Chien, M. T.; Nakazato, H., Joint numerical range and its generating hypersurface, Linear Algebra Appl., 432, 173-179 (2010) · Zbl 1185.15021 [5] Fiedler, M., Geometry of the numerical range of matrices, Linear Algebra Appl., 37, 81-96 (1981) · Zbl 0452.15024 [6] Helton, J. W.; Vinnikov, V., Linear matrix inequality representations of sets, Comm. Pure Appl. Math., 60, 654-674 (2007) · Zbl 1116.15016 [7] Henrion, D., Detecting rigid convexity of bivariate polynomial, Linear Algebra Appl., 432, 1218-1233 (2010) · Zbl 1183.65023 [8] Kippenhahn, R., Über den wertevorrat einer Matrix, Math. Nachr., 6, 193-228 (1951) · Zbl 0044.16201 [9] Lax, P. D., Differential equations, difference equations and matrix theory, Comm. Pure Appl. Math., 6, 175-194 (1958) · Zbl 0086.01603 [10] Lewis, A. S.; Parrilo, P. A.; Ramana, M. V., The Lax conjecture is true, Proc. Amer. Math. Soc., 133, 2495-2499 (2005) · Zbl 1073.90029 [11] Li, C. K., C-numerical ranges and C-numerical radii, Linear and Multilinear Algebra, 37, 51-82 (1994) · Zbl 0814.15022 [12] Li, C. K.; Poon, Y. T., Some results on the $$c$$-numerical range, (Five Decades as a Mathematician and Educator (1995), World Sci. Publ.: World Sci. Publ. River Edge), 247-258 · Zbl 0902.47002 [13] Liess, O., Remarks on the Lax conjecture for hyperbolic polynomials, Linear Algebra Appl., 430, 2123-2132 (2009) · Zbl 1211.15010 [14] Vinnikov, V., Self-adjoint determinantal representations of real plane curves, Math. Ann., 296, 453-479 (1993) · Zbl 0789.14029 [15] Westwick, R., A theorem on numerical range, Linear and Multilinear Algebra, 2, 311-315 (1975) · Zbl 0303.47001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.