On the Matrix Polynomial, Lambda-Matrix and Block Eigenvalue Problems
Abstract
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.
Document Details
- Document Type
- Technical Report
- Publication Date
- Dec 17, 1971
- Accession Number
- AD0735478
Entities
People
- J. E. Dennis Jr.
- Joseph F. Traub
- R. P. Weber
Organizations
- Carnegie Mellon University