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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Groups of banded matrices with banded inverses


Author: Gilbert Strang
Journal: Proc. Amer. Math. Soc. 139 (2011), 4255-4264
MSC (2010): Primary 15A23
Published electronically: April 29, 2011
MathSciNet review: 2823071
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Abstract: A product $ A=F_1 \ldots F_N$ of invertible block-diagonal matrices will be banded with a banded inverse: $ A_ij=0$ and also $ (A^{-1})_{ij}=0$ for $ \vert i-j\vert>w$. We establish this factorization with the number $ N$ controlled by the bandwidths $ w$ and not by the matrix size $ n.$ When $ A$ is an orthogonal matrix, or a permutation, or banded plus finite rank, the factors $ F_i$ have $ w=1$ and we find generators of that corresponding group. In the case of infinite matrices, the $ A=LPU$ factorization is now established but conjectures remain open.


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Additional Information

Gilbert Strang
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: gs@math.mit.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10959-6
PII: S 0002-9939(2011)10959-6
Keywords: Banded matrix, banded inverse, Bruhat permutation, factorization, group generators, shifting index
Received by editor(s): October 22, 2010
Published electronically: April 29, 2011
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.