On the Matrix Polynomial, Lambda-matrix and Block Eigenvalue ProblemsA matrix S is a solvent of the matrix polynomial M(X) identically equal to X sup m + A(sub 1) X sup(M - 1) + ... + A sub m, if M(S) = 0, where A sub i, X and S are square matrices. The authors present some new mathematical results for matrix polynomials, as well as a globally convergent algorithm for calculating such solvents. In the theoretical part of this paper, existence theorems for solvents, a generalized division, interpolation, a block Vandermonde, and a generalized Lagrangian basis are studied. Algorithms are presented which generalize Traub's scalar polynomial methods, Bernoulli's method, and eigenvector powering. The related lambda-matrix problem, that of finding a scalar lambda such that I(lambda sup m) + A(sub 1)lambda sup(M - 1) + ... + A sup m is singular, is examined along with the matrix polynomial problem. The matrix polynomial problem can be cast into a block eigenvalue formulation as follows. Given a matrix A of order mn, find a matrix X of order n, such that AV = VX, where V is a matrix of full rank. Some of the implications of this new block eigenvalue formulation are considered. |
Contents
CHAPTER III | 29 |
The Block Vandermonde | 36 |
A Matrix Polynomial Algorithm | 44 |
1 other sections not shown
Common terms and phrases
A₁ A₂ Algorithm Bell Telephone Laboratories Bernoulli iteration Bernoulli method Bézout's Theorem block companion matrix block eigen block eigenvalue problem block eigenvector block matrix block Vandermonde block vector Chapter complete set computations considered CORN CORNELL UNIVERSITY Definition distinct latent roots dominant latent root dominant solvent doold eigenvector powering equation exists a unique full rank fundamental matrix polynomials given Hoold Jordan form lambda lambda-matrix problem lambda-vector Lancaster leading matrix coefficient left solvent Lemma LIBRARY UNIVERSITY linearly independent locally convergent M₁ M₁(X matrix poly matrix polynomial M(X matrix polynomial problem meldong monic matrix polynomial Newton's method nomial nonsingular nonsingular matrix order mn P₁ Proof R₁ right solvents roots of M(1 S₁ S₂ set of block set of right set of solvents singular solvent of M(X Theorem 5.1 Traub's scalar polynomial UNIV CORNELL V(S₁ V₁ Vandermonde matrix VERSITY X₁ Ιλ