Cubic surfaces and \(q\)-numerical ranges (Q2862110)

Let \(A\) be an \(n\times n\) matrix with complex entries and \(0\leq q\leq 1\). The \(q\)--numerical range of \(A\) is defined by \(W_q(A)=\{x^\ast A y, \;x,y \in\mathbb{C}^n, x^\ast x=y^\ast y=1, x^\ast y=q\}\). Here \(x^\ast\) denotes the transpose of the coordinate--wise complex conjugate of \(x\). \(W_q(A)\) is a convex subset of \(\mathbb{C}\).NEWLINENEWLINEThe boundary of \(W_q(A)\) is the orthogonal projection of a hypersurface defined by the dual surface of the homogeneous polynomial \(F(t,x,y,z)=\det (tI_n+x(A+A^\ast)/2+y(A-A^\ast/2i+zA^\ast A)\).NEWLINENEWLINEDifferent types of cubic surfaces corresponding to \(F\) for some \(3\times 3\) matrices are studied.