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The Algebra of Elimination

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Excursions in Harmonic Analysis, Volume 3

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

Elimination with only the necessary row exchanges will produce the triangular factorization A = LPU, with the (unique) permutation P in the middle. The entries in L are reordered in comparison with the more familiar \(A={\widehat{P}\widehat{L}\widehat{U}}\) (where \({\widehat{P}}\) is not unique). Elimination with three other starting points 1, n and n, n and n, 1 produces three more factorizations of A, including the Wiener–Hopf form UPL and Bruhat’s \(U_1\,\pi\,U_2\) with two upper triangular factors.

All these starting points are useless for doubly infinite matrices. The matrix has no first or last entry. When A is banded and invertible, we look for a new way to establish A = LPU. The key is to locate the pivot rows (we also find the main diagonal of A). LPU was previously known in the periodic (block Toeplitz) case \(A(i,j)=A(i+b,j+b)\), by factoring a matrix polynomial.

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Acknowledgements

This chapter was begun on a visit to the beautiful Hong Kong University of Science and Technology. We thank its faculty and its president Tony Chan for their warm welcome. On my final morning, George Lusztig happened to walk by—and he clarified the history of Bruhat. The books of Marko Lindner (and our email correspondence) introduced me to the limit analysis of non-Toeplitz infinite matrices.

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Authors and Affiliations

  1. Massachusetts Institute of Technology, Cambridge, MA, USA

    Gilbert Strang

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  1. Gilbert Strang
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Correspondence to Gilbert Strang .

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Editors and Affiliations

  1. Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, Maryland, USA

    Radu Balan

  2. Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, Maryland, USA

    Matthew J. Begué

  3. Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, Maryland, USA

    John J. Benedetto

  4. Department of Mathematics, University of Maryland, Norbert Weiner Center, College Park, Maryland, USA

    Wojciech Czaja

  5. Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, Maryland, USA

    Kasso A. Okoudjou

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Strang, G. (2015). The Algebra of Elimination. In: Balan, R., Begué, M., Benedetto, J., Czaja, W., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 3. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-13230-3_1

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