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Invariant Factors Under Rank One Perturbations

Published online by Cambridge University Press:  20 November 2018

Robert C. Thompson
Robert C. Thompson*
Affiliation:
University of California, Santa Barbara, California
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Let R be a principal ideal domain, i.e., a commutative ring without zero divisors in which every ideal is principal. The invariant factors of a matrix A with entries in R are the diagonal elements when A is converted to a diagonal form D = UAV, where U, V have entries in R and are unimodular (invertible over R), and the diagonal entries d1 …, dn of D form a divisibility chain: d1|d2| … |dn. Very little has been proved about how invariant factors may change when matrices are added. This is in contrast to the corresponding question for matrix multiplication, where much information is now available [6].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 1 , 01 February 1980 , pp. 240 - 245
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. De Oliveira, G. N., Marques de Sa, E., and Dias daSilva, J. A., On the eigenvalues of the matrix A + XBX∼l , Linear Multilinear Algebra, in press.Google Scholar
2. Gantmacher, F. R., Theory of matrices, second edition, Moscow (1966), page 148.Google Scholar
3. Marques de Sa, E., Imbedding conditions for \-matrices, Linear Algebra and Applications, in press.Google Scholar
4. Thompson, R. C., Interlacing inequalities for invariant factors, Linear Algebra and Applications, to appear.Google Scholar
5. Thompson, R. C., The behavior of eigenvalues and singular values under perturbations of restricted rank, Linear Algebra and Applications. 13 (1976), 69–78.Google Scholar
6. Thompson, R. C., in preparation.Google Scholar
7. Thompson, R. C., in preparation.Google Scholar