Bidiagonal matrix: Difference between revisions
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In [[mathematics]], a '''bidiagonal matrix''' is a [[banded matrix]] with non-zero entries along the main diagonal and ''either'' the diagonal above or the diagonal below. This means there are exactly two non |
In [[mathematics]], a '''bidiagonal matrix''' is a [[banded matrix]] with non-zero entries along the main diagonal and ''either'' the diagonal above or the diagonal below. This means there are exactly two non-zero diagonals in the matrix. |
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When the diagonal above the main diagonal has the non-zero entries the matrix is '''upper bidiagonal'''. When the diagonal below the main diagonal has the non-zero entries the matrix is '''lower bidiagonal'''. |
When the diagonal above the main diagonal has the non-zero entries the matrix is '''upper bidiagonal'''. When the diagonal below the main diagonal has the non-zero entries the matrix is '''lower bidiagonal'''. |
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* [[List of matrices]] |
* [[List of matrices]] |
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* [[LAPACK]] |
* [[LAPACK]] |
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* [[Hessenberg form]] The Hessenberg form is similar, but has more non |
* [[Hessenberg form]] The Hessenberg form is similar, but has more non-zero diagonal lines than 2. |
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==References== |
==References== |
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Revision as of 02:00, 25 February 2022
In mathematics, a bidiagonal matrix is a banded matrix with non-zero entries along the main diagonal and either the diagonal above or the diagonal below. This means there are exactly two non-zero diagonals in the matrix.
When the diagonal above the main diagonal has the non-zero entries the matrix is upper bidiagonal. When the diagonal below the main diagonal has the non-zero entries the matrix is lower bidiagonal.
For example, the following matrix is upper bidiagonal:
and the following matrix is lower bidiagonal:
Usage
One variant of the QR algorithm starts with reducing a general matrix into a bidiagonal one,[1] and the Singular value decomposition uses this method as well.
Bidiagonalization
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See also
- List of matrices
- LAPACK
- Hessenberg form The Hessenberg form is similar, but has more non-zero diagonal lines than 2.
References
- Stewart, G. W. (2001) Matrix Algorithms, Volume II: Eigensystems. Society for Industrial and Applied Mathematics. ISBN 0-89871-503-2.
- ^ Bochkanov Sergey Anatolyevich. ALGLIB User Guide - General Matrix operations - Singular value decomposition . ALGLIB Project. 2010-12-11. URL:http://www.alglib.net/matrixops/general/svd.php. Accessed: 2010-12-11. (Archived by WebCite at https://www.webcitation.org/5utO4iSnR)
External links
- High performance algorithms for reduction to condensed (Hessenberg, tridiagonal, bidiagonal) form
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