The primary objective of this article is to demonstrate that Rayleigh’s quotient and its variants retain the usual properties of boundedness and stationarity even when the linear vibratory system is non-classically damped, extending previously accepted results that these quotients could attain stationarity when damping was proportional or the modal damping matrix was diagonally dominant. This conclusion is reached by allowing the quotients to be defined in complex space and using complex differentiation. A secondary objective is to show how these quotients and their associated eigenvalue problems can be combined to generate bounds on the system’s eigenvalues, an immediate consequence that follows from establishing boundedness and stationarity in complex space. The reported bounds are simple to compute and appear to be tighter than previous bounds reported in the literature.

We consider the class of real second-order linear dynamical systems that admit real diagonal forms with the same eigenvalues and partial multiplicities. The nonzero leading coefficient is allowed to be singular, and the associated quadratic matrix polynomial is assumed to be regular. We present a method and algorithm for converting any such n-dimensional system into a set of n mutually independent second-, first-, and zeroth-order equations. The solutions of these two systems are related by a real, time-dependent, and nonlinear n-dimensional transformation. Explicit formulas for computing the 2 n× 2 n real and time-invariant equivalence transformation that enables this conversion are provided. This paper constitutes a complete solution to the problem of diagonalizing a second-order linear system while preserving its associated Jordan canonical form.