On the Matrix Polynomial, Lambda-Matrix and Block Eigenvalue Problems
CARNEGIE-MELLON UNIV PITTSBURGH PA DEPT OF COMPUTER SCIENCE
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A matrix S is a solvent of the matrix polynomial MX identically equal to X sup m Asub 1 X supM - 1 ... A sub m, if MS 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 Traubs scalar polynomial methods, Bernoullis method, and eigenvector powering. The related lambda-matrix problem, that of finding a scalar lambda such that Ilambda sup m Asub 1lambda supM - 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.
- Theoretical Mathematics