Joint numerical range and its generating hypersurface
The authors consider the joint numerical range \(W(H_1, \dots, H_n)\) of \(m\)-tuples of Hermitian matrices. Associated with an~\(m\)-tuple \(H_1, \dots, H_n\) there is the homogeneous polynomial \[ F(y_0,\dots, y_m) = \det (y_0I_n+ y_1H_1 + \dots + y_mH_m), \] which may be looked upon as a~hyperbolic form with respect to the point \((1, 0, 0, \dots, 0) \in{\mathbb R}^{m+1}\). Consider the algebraic variety \[ S_F=\{[(y_0,y_1,\dots, y_m)]\in {\mathbb C\mathbb P}^m: F(y_0,y_1,\dots, y_m)= 0\}. \] The main purpose of this paper is to examine the joint numerical in terms of the hypersurface \(S_F\) and the boundary generating hypersurface \(S_F^\land\). An~example is presented to show that an~analog of Kippenhahn's theorem for the joint numerical range of three Hermitian matrices does not hold.
scientific article; zbMATH DE number 5650029
scientific article