Davis-Wielandt shell and \(q\)-numerical range (Q5956237)

The purpose of the paper is to describe the boundary of the \(q\)-numerical range \(W_q(A)=\{x^*Ay\): \(x,y\in{\mathbb C}^n\), \(|x|=|y|=1\), \(x^*y=q\}\), \(q\in {\mathbb C}\), \(|q|\leqslant 1\), of a square \(n\times n\) complex matrix \(A\) by involving its Davis-Wielandt shell \(DW(A)=\{(x^*Ax, x^*A^*Ax)\): \(x\in {\mathbb C}^n\), \(|x|=1\}\), which is closely related with \(W_q(A)\) as it is shown by \textit{C.-K.~Li} and \textit{H.~Nakazato} [Linear Multilinear Algebra 43, No. 4, 385-409 (1998; Zbl 0899.15017)]. The obtained result is used to generate an algorithm for plotting the \(q\)-numerical range of \(A\). For a special class of matrices their \(q\)-numerical ranges are completed by using this method.