The boundary of higher rank numerical ranges
For an \(n\)-by-\(n\) complex matrix \(A\) and a positive integer \(k\), \(1\leq k\leq n\), the rank-\(k\) numerical range \(\Lambda_k(A)\) of \(A\) is defined as the subset \(\{\lambda\in\mathbb{C}:PAP= \lambda P\) for some rank-\(k\) orthogonal projection \(P\}\) of the complex plane. It is known that \(\Lambda_k(A)\) is a compact convex set and may be empty. When \(k= 1\), \(\Lambda_1(A)\) is exactly the classical numerical range \(W(A)\). The present paper is devoted to the study of the boundary curve of \(\lambda_k(A)\) via the homogeneous form \[ F_A(t,x,y)= \text{det}(tI_n+ x\cdot\text{Re\,}A+ y\cdot\text{Im\,}A) \] associated with \(A\), where \(\text{Re\,}A= (A+ A^*)/2\) and \(\text{Im\,}A= (A- A^*)/(2i)\) are the real and imaginary parts of \(A\), respectively. In Section 2, the authors propose a method to generate \(\partial\Lambda_k(A)\) via symbolic calculations and analysis of the singularities of the associated curve. They illustrate this by a 4-by-4 real tridiagonal matrix and a 6-by-6 complex symmetric matrix. Then, in Section 3, they consider an explicit \(2m\)-by-\(2m\) \((m\geq 2)\) matrix \(A\) with the algebraic curve \(F_A(1,x,y)= 0\) given by the roulette curve \[ z(s)= \exp(-i(m- 1)s)+ a\exp(ims)\quad\text{for }0\leq s\leq 2\pi, \] where \(a> 1\). It is shown that the total number of line segments on \(\partial\Lambda_k(A)\) for \(1\leq k\leq m\) is \((2m- 1)(m- 1)\). This shows that the upper bound \((2m- 1)(m- 1)\) for such numbers of \(2m\)-by-\(2m\) matrices is attained by such an \(A\). Finally, in Section 4, the authors give a 4-by-4 matrix \(A\) with \(\Lambda_2(A)\) not equal to the classical numerical range \(W(B)\) of any matrix \(B\). This is in sharp contrast to the case for \(c\)-numerical ranges. It concludes with a sufficient condition that the \(n\)-by-\(n\) matrix \(A\) is such that \(\partial\Lambda_k(A)\) \((k\geq 2)\) has sharp points and \(F_A(t,x,y)\) is irreducible for the nonattainability of \(\Lambda_k(A)\) by any \(W(B)\).
scientific article; zbMATH DE number 5943586
scientific article