The Interplay of Ranks of Submatrices
Name
HPCES009.pdf
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117.23 KB
Format
Adobe PDF
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5765705940153fa162f6811e1ceeb373
Author(s)
Strang, Gilbert
Nguyen, Tri Dung
Date Issued
January 2004
Series/Report no.
High Performance Computation for Engineered Systems (HPCES);
Abstract
A banded invertible matrix T has a remarkable inverse. All "upper" and "lower" submatrices of T⁻¹ have low rank (depending on the bandwidth in T). The exact rank condition is known, and it allows fast multiplication by full matrices that arise in the boundary element method.
We look for the "right" proof of this property of T⁻¹. Ultimately it reduces to a fact that deserves to be better known: Complementary submatrices of any T and T⁻¹ have the same nullity. The last figure in the paper (when T is tridiagonal) shows two submatrices with the same nullity n – 3. Then C has rank 1. On and above the diagonal of T⁻¹, all rows are proportional.
Subjects
band matrix
low rank submatrix
fast multiplication.
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