LXXV. The numerical reduction of non-singular matrix pencils

Publisher Website
Google Scholar
Abstract
The problem of reducing a non-singular matrix pencil L≡l 1 L 1+l 2 L 2, |L|≢0, to canonical form, and of finding the latent roots l 1 : l 2 and root- vectors x that satisfy Lx=0, arises in many branches of applied mathematics. The most familiar of these applications is the analysis of small vibrations of a dynamical system about an equilibrium position by means of Lagrangian frequency equations ; while other applications include the conditioned stationary values of multivariate functions, approximations to certain integral equations, orthogonal mean square regressions in multivariate statistics, linear operational equations with constant coefficients, the determination of matrix commutants, matrix interpolation, and the growth behaviour of animal populations. The stock of current reduction procedures embraces the determinantal method, the Duncan-Collar-Aitken- Hotelling method, the Morris escalator, Samuel-son's method, Horst's method, and the Cayley-Hamilton method. For one reason or another, each of these methods proves somewhat unsatisfactory—either it demands considerable cpmputing labour, or it is unsuitable for automatic computors, or it only solves certain sections of the problem, or it is restricted to special cases such as symmetric matrices or distinct latent roots. In this paper I propound a new method, which I consider an improvement on current procedures. This new method is an extension of the Cayley-Hamilton method, based upon the general theory of collineations. It is quite general, not requiring modification in special cases ; it provides the complete solution of all the latent roots and all the root-vectors and/or invariant space bases ; it is suitable for punched card analysis and automatic computors—with the exception of a simple set of divisions, ring operations alone suffice for the whole computation—and it calls for rather less computing labour than current methods.

This publication has 22 references indexed in Scilit: