On the Matrix Polynomial, Lambda-matrix and Block Eigenvalue Problems

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Cornell University, 1971 - Eigenvalues - 234 pages
A 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.

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Contents

CHAPTER III
29
The Block Vandermonde
36
A Matrix Polynomial Algorithm
44

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