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The Interplay of Ranks of Submatrices

Author(s)
Strang, Gilbert; Nguyen, Tri Dung
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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.
Date issued
2004-01
URI
http://hdl.handle.net/1721.1/3885
Series/Report no.
High Performance Computation for Engineered Systems (HPCES);
Keywords
band matrix, low rank submatrix, fast multiplication.

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