Groups of banded matrices with banded inverses
Gilbert Strang
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Source: Crossref
Published: Apr 29, 2011
DOI: 10.1090/s0002-9939-2011-10959-6
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A product A = F 1 … F N A=F_1\,\ldots \,F_N of invertible block-diagonal matrices will be banded with a banded inverse: A i j = 0 A_ij=0 and also ( A − 1 ) i j = 0 (A^{-1})_{ij}=0 for | i − j | > w |i-j|>w . We establish this factorization with the number N N controlled by the bandwidths w w and not by the matrix size n . n. When A A is an orthogonal matrix, or a permutation, or banded plus finite rank, the factors F i F_i have w = 1 w=1 and we find generators of that corresponding group. In the case of infinite matrices, the A = L P U A=LPU factorization is now established but conjectures remain open.
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