Further bounds for the smallest singular value and the spectral condition number (Q1972510)

Let \({\mathbf A}\) be an \(n\times n\) complex matrix, and let \(\sigma _1({\mathbf A})\geq \sigma _2({\mathbf A})\geq\dots\geq \sigma _n({\mathbf A})\) be the singular values of \({\mathbf A}\). The author shows how to construct an increasing sequence of lower bounds for \(\sigma _n({\mathbf A})\) which improves the bounds of \textit{Y. Yu} and \textit{D. Gu} [Linear Algebra Appl. 253, 25-38 (1997; Zbl 0876.15015)]. The spectral condition number \(\kappa _2({\mathbf A})= \sigma _1({\mathbf A})/\sigma _n({\mathbf A})\) measures the sensibility of the solution of \({\mathbf A}{\mathbf x}={\mathbf b}\) to errors in the data or to round-off errors, and one can estimate \(\kappa _2({\mathbf A})\) using a lower bound for \(\sigma _n({\mathbf A})\) and an upper bound for \(\sigma _1({\mathbf A})\). New upper bounds for \(\kappa _2({\mathbf A})\) are derived in the last part of the article.