The \(c\)-numerical range of a rank one matrix (Q5938897)

Let \(a,b\in {\mathbb C}^n\), and let \(c=(c_1,\dots, c_n)\in {\mathbb R}^n\) with \(c_1\geq c_2\geq\dots\geq c_n\). Then the \(c\)-numerical range \(W_c(A)\) of a rank one matrix \(A=ab^*\) is an elliptic disc with foci \(c_nb^*a\) and \(c_1b^*a\) and semiminor axis \(\frac 12 (c_1-c_2)(\|a\|^2\|b\|^2-|b^*a|^2)^{1/2}\).