The Interplay of Ranks of Submatrices
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....
| Main Authors: | , |
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| Format: | Article |
| Language: | en_US |
| Published: |
2003
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| Subjects: | |
| Online Access: | http://hdl.handle.net/1721.1/3885 |
| _version_ | 1826201228893224960 |
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| author | Strang, Gilbert Nguyen, Tri Dung |
| author_facet | Strang, Gilbert Nguyen, Tri Dung |
| author_sort | Strang, Gilbert |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T11:48:14Z |
| format | Article |
| id | mit-1721.1/3885 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T11:48:14Z |
| publishDate | 2003 |
| record_format | dspace |
| spelling | mit-1721.1/38852019-04-10T18:39:32Z The Interplay of Ranks of Submatrices Strang, Gilbert Nguyen, Tri Dung band matrix low rank submatrix fast multiplication. 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. Singapore-MIT Alliance (SMA) 2003-12-14T22:43:24Z 2003-12-14T22:43:24Z 2004-01 Article http://hdl.handle.net/1721.1/3885 en_US High Performance Computation for Engineered Systems (HPCES); 120039 bytes application/pdf application/pdf |
| spellingShingle | band matrix low rank submatrix fast multiplication. Strang, Gilbert Nguyen, Tri Dung The Interplay of Ranks of Submatrices |
| title | The Interplay of Ranks of Submatrices |
| title_full | The Interplay of Ranks of Submatrices |
| title_fullStr | The Interplay of Ranks of Submatrices |
| title_full_unstemmed | The Interplay of Ranks of Submatrices |
| title_short | The Interplay of Ranks of Submatrices |
| title_sort | interplay of ranks of submatrices |
| topic | band matrix low rank submatrix fast multiplication. |
| url | http://hdl.handle.net/1721.1/3885 |
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