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On the Matrix Polynomial, Lamda-Matrix and Block Eigenvalue Problems

dc.contributor.authorDennis, John E., Jr.en_US
dc.contributor.authorTraub, J.F.en_US
dc.contributor.authorWeber, R.P.en_US
dc.date.accessioned2007-04-19T17:56:16Z
dc.date.available2007-04-19T17:56:16Z
dc.date.issued1971-09en_US
dc.description.abstractA matrix $S$ is a solvent of the matrix polynomial $M(X) \equiv X^{m} + A_{1}X^{m-1} + \cdots + A_{m}$, if $M(S) = \stackrel{0}{=}$, where $A_{1}, X$ and $S$ are square matrices. We 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 $I \lambda^{m} + A_{1} \lambda^{m-1} + \cdots + A_{m}$ 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.en_US
dc.format.extent4207450 bytes
dc.format.extent3115144 bytes
dc.format.mimetypeapplication/pdf
dc.format.mimetypeapplication/postscript
dc.identifier.citationhttp://techreports.library.cornell.edu:8081/Dienst/UI/1.0/Display/cul.cs/TR71-109en_US
dc.identifier.urihttps://hdl.handle.net/1813/5954
dc.language.isoen_USen_US
dc.publisherCornell Universityen_US
dc.subjectcomputer scienceen_US
dc.subjecttechnical reporten_US
dc.titleOn the Matrix Polynomial, Lamda-Matrix and Block Eigenvalue Problemsen_US
dc.typetechnical reporten_US

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