Generalized Rayleigh-quotient formulas for the eigenvalues of self-adjoint matrices. (English) Zbl 1361.15010
The author derives analogues of the classical min-max and max-min characterizations of the eigenvalues of an \(n\times n\) Hermitian matrix \(A\). In these analogues, the Rayleigh quotient is replaced by \((Au,v)/(u,v)\), with \((u,v)>0\), where \(u\), \(v\) belong to a certain subspace of \({\mathbb C}^n\) isomorphic to \({\mathbb R}^n\). Corresponding results for singular values of a general matrix are illustrated by a numerical example arising from the description of a simple damped vibrating system.
Reviewer: Alan L. Andrew (Bundoora)
MSC:
15A18 | Eigenvalues, singular values, and eigenvectors |
15A42 | Inequalities involving eigenvalues and eigenvectors |
15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
15B57 | Hermitian, skew-Hermitian, and related matrices |