Numerical methods for finding characteristic roots and vectors of matrices
Wilfred M. Kincaid · Quarterly of Applied Mathematics · 1947
The present paper treats the problem of finding the characteristic roots and vectors of a matrix (having linear elementary divisors).The emphasis throughout is on methods of getting numerical results in practical cases rather than on theoretical questions.Symmetric matrices are taken up first, and methods are discussed for finding (a) the largest characteristic root and corresponding vector, and (b) the other roots and vectors.All these methods are variants of the iteration process.They are then extended to general matrices, with particular reference to the case of complex roots.The paper closes with a brief discussion of the solution of algebraic equations by means of matrices.Among earlier work on this subject, we may mention that of Hotelling1 and Aitken.2Indeed, much of the material in the present paper is taken from Aitken's, through some modifications have been introduced, particularly with regard to the determination of roots other than the largest.A recent paper by Fry3 takes up matrices in connection with the solution of algebraic equations.The present paper, particularly the last section, is thus in a measure supplementary to Fry's.Wayland,4 on the other hand, gives methods for reducing the problem of finding the roots of a given matrix to that of solving an algebraic equation; this matter is touched upon Sec.II.Recent work along these lines has also been done by Morris6 and Head.6At this point the author wishes to acknowledge his indebtedness to Prof.