The Generalized Rayleigh Quotient

Abstract

In this paper we generalize the concept of the Rayleigh quotient to a complex Banach space. Lord Rayleigh investigated the quotient(1)considered as a function of the components of q, in the case of a symmetric matrix pencil Aλ+C with A positive definite. It is known that R(q) has a stationary value when q is a characteristic vector of Aλ+C and that(2)where qi is a characteristic vector corresponding to the characteristic value λi