Elliptic numerical ranges of bordered matrices (Q439197)

For an \(n\times n\) complex matrix \(A\), its numerical range \(W(A)\) is the subset \(\{\langle Ax,x\rangle: x\in\mathbb{C}^n\), \(\| x\|= 1\}\) of the complex plane, where \(\langle.,.\rangle\) and \(\|.\|\) denote, respectively, the standard inner product and its associated norm in \(\mathbb{C}^n\).NEWLINENEWLINE The present paper starts with a criterion for \(W(A)\) to be an elliptic disc. The criterion is in terms of the maximum eigenvalue of the matrix \((e^{i\theta} A+\overline e^{i\theta} A^*)/2\) for real \(\theta\). It is an imitation of the one for \(W(A)\) to be a circular disc given by Marcus and Pesce.NEWLINENEWLINE The authors then apply the criterion to show that the numerical range of a bordered matrix of a scalar matrix \([a_{ij}]^n_{i,j=1}\), where \(a_{ij}=\alpha\) if \(1\leq i= j\leq n-1\), and \(0\) if \(1\leq i\neq j\leq n-1\), is an elliptic disc, which is first obtained by Linden. It is also shown that certain \(n\times n\) bordered matrices of the form NEWLINE\[NEWLINE\left[\begin{matrix} \alpha I_k & C\\ D &\beta I_{n-k}\end{matrix}\right]NEWLINE\]NEWLINE also have elliptic disc numerical ranges.