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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.

References [Enhancements On Off] (What's this?)

  • [1] C. Albert, C.-K. Li, G. Strang and G. Yu, Permutations as products of parallel transpositions, submitted to SIAM J. Discrete Math. (2010).
  • [2] E. Asplund, Inverses of matrices $ \{a_{ij}\}$ which satisfy $ a_{ij}=0$ for $ j>i+p$, Math. Scand. 7 (1959) 57-60. MR 0109833 (22:718)
  • [3] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988) 909-996. MR 951745 (90m:42039)
  • [4] I. Gohberg, M. Kaashoek, and I. Spitkovsky. An overview of matrix factorization theory and operator applications, Operator Th. Adv. Appl., Birkhäuser (2003) 1-102. MR 2021095 (2004j:47034)
  • [5] V. Olshevsky, P. Zhlobich, and G. Strang, Green's matrices, Lin. Alg. Appl. 432 (2010) 218-241. MR 2566471 (2011a:15017)
  • [6] G. Panova, Factorization of banded permutations, (2010).
  • [7] J. Plemelj, Riemannsche Funktionenscharen mit gegebener Monodromiegruppe, Monat. Math. Phys. 19 (1908) 211-245. MR 1547764
  • [8] V.S.G. Raghavan, Banded Matrices with Banded Inverses, M.Sc. Thesis, MIT (2010).
  • [9] M.D. Samson and M.F. Ezerman, Factoring permutation matrices into a product of tridiagonal matrices, (2010).
  • [10] G. Strang, Fast transforms: Banded matrices with banded inverses, Proc. Natl. Acad. Sci. 107 (2010) 12413-12416. MR 2670987
  • [11] G. Strang, Banded matrices with banded inverses and A = LPU, Proceedings of ICCM2010 (International Congress of Chinese Mathematicians, Beijing, December 2010).
  • [12] G. Strang and T. Nguyen, The interplay of ranks of submatrices, SIAM Review 46 (2004) 637-646. MR 2124679 (2005m:15015)
  • [13] K. TeKolste, private communication (2010).

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Additional Information

Gilbert Strang
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

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.

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