Related bounds for the extreme eigenvalues (Q1972511)

Let \({\mathbf A}\) be a matrix with real eigenvalues \(\lambda _1\geq \lambda _2\geq ...\geq \lambda _n\). The authors present a method to construct an increasing (respectively, decreasing) sequence of lower (respectively, upper) bounds for \(\lambda _n\) (respectively, \(\lambda _1\)) in the case where \(\lambda _1\) (respectively, \(\lambda _n\)) is known. This situation occurs for example if \({\mathbf A}\) is an irreducible stochastic matrix with real spectrum, and in this case \(\lambda _1=1\). A method to construct a decreasing sequence of upper bounds for the largest singular value of an \(m\times n\) complex matrix is obtained as an application of the main result.