XX.—Studies in Practical Mathematics. II. The Evaluation of the Latent Roots and Latent Vectors of a Matrix

Abstract

In many branches of applied mathematics problems arise which require for their solution a knowledge of the latent roots of a matrix A, sometimes only the root of greatest modulus but often the second and other roots as well, and the corresponding latent vectors. A few examples, among many that might be cited, are problems in the dynamical theory of oscillations, problems of conditioned maxima and minima, problems of correlation between statistical variables, the determination of the principal axes of quadrics, and the solution of differential or other operational equations. It is important, therefore, to have a choice of methods for obtaining latent roots and latent vectors without undue labour, and the object of the present paper is to augment the existing store of such methods.

Sign wave analysis in matrix eigenvalue problems

“…The power method ( [1]; [3], p. 296; [5]; [7]; [9]; [10]) is generally recognized as a numerically efficient algorithm for determining the dominant eigenvalue (s) and associated eigenvector(s) of a matrix. We review the method briefly for the case where the matrix A has a single dominant eigenvalue X with associated eigenvector u.…”

“…Denoting by the superscript T the transpose of a vector or matrix, we let v be an eigenvector of A T associated with the eigenvalue X. Starting with any vector xm) satisfying uTa;<0> ^ 0, we now form by successive matrix-vector multiplications the vectors The convergence of the process can be sped up by devices such as shift of the origin [10], fractional iteration [8], and the 5 -process [1]. Statements similar to the above still hold if the multiplicity of X is greater than one, but the convergence may then be slow due to the presence of nonlinear divisors.…”

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“…Three basic techniques for solution of determinantal equations must be considered: (1) Direct solution of the equation by numerical methods [l]. (2) Direct solution by the method of matrix multiplication [2][3][4][5][6] (directly applicable only to the case | A -A| =0). (3) Expansion into polynomial form [6][7][8][9][10][11][12][13][14] and solution of the polynomial equation by standard methods.…”

“…II,No. 4 (l/3)(«3 + 2n -3) M-D (multiplications and divisions), (w/6)(» + 1)(2» + 1) A-S (additions and subtractions).…”

“…Hence if k^n+1 the expansion to polynomial form represents a net saving, still using the powerful iterative method [3,4,16,17,18].…”