An optimal bound for the spectral variation of two matrices (Q1070007)

Let \(A\) and \(B\) be \(n\times n\) matrices with eigenvalues \(\{\lambda_ 1,\ldots,\lambda_ n\}\) and \(\{\mu_ 1,\ldots,\mu_ n\}\), respectively. Let \(s_ A(B)=\max_{j}\min_{i}| \lambda_ i- \mu_ j|\), and \(\| A\| =\sup \| Ax\| /\| x\|\) where \(\| x\|\) is the Euclidean vector norm. By using Hadamard's inequality the author proves that \(s_ A(B)\leq (\| A\| +\| B\|)^{1-1/n}\| A-B\|^{1/n}\) and characterizes the cases when equality holds. The rate of continuity of the spectrum was studied by \textit{B. Aupetit} and the reviewer [Linear Algebra Appl. 52/53, 39-44 (1983; Zbl 0518.46035)] where precise exponents of \(\| A-B\|\) were obtained (locally).