On eigenvalues and boundary curvature of the numerical range
For an \(n\times n\) matrix \(A\) let \(W(A)\) be its numerical range, and \(\sigma (A)\) be its spectrum. By the theorem of \textit{W. F. Donoghue jun.} [Michigan Math. J. 4, 261-263 (1958; Zbl 0082.11601)], all corner points of \(W(A)\) belong to~\(\sigma (A)\). The authors obtain bounds for the distance \(D_p(A)\) from a point~\(p\) on the boundary \(\partial W(A)\) to \( \sigma (A)\). Let \(M(A)\) be the smallest constant such that \(D_p(A)\leqslant M(A)R_p(A)\) for all \(p\in \partial W(A)\), where \(R_p(A)\) is the radius of curvature of~\(\partial W(A)\), being defined for all but finitely many points, and let \(M_n=\sup \{M(A) : A\in {\mathbb C}^{n\times n}\}\). The main result is the following inequality: \(\frac n2\sin \frac \pi n\leqslant M_n\leqslant \frac n2\), which becomes an identity for \(n=2\). A description of some matrices \(A\) where \(M(A)=1\), as well as some computations of exact values of \(M(A)\) for \(2\times 2\) matrices \(A\) is given.
scientific article; zbMATH DE number 1559345