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<div>In [[mathematics]], a '''bidiagonal matrix''' is a [[matrix (mathematics)|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.<br />
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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'''.<br />
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For example, the following matrix is '''upper bidiagonal''':<br />
:<math>\begin{pmatrix}<br />
1 & 4 & 0 & 0 \\<br />
0 & 4 & 1 & 0 \\<br />
0 & 0 & 3 & 4 \\<br />
0 & 0 & 0 & 3 \\<br />
\end{pmatrix}</math><br />
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and the following matrix is '''lower bidiagonal''':<br />
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:<math>\begin{pmatrix}<br />
1 & 0 & 0 & 0 \\<br />
2 & 4 & 0 & 0 \\<br />
0 & 3 & 3 & 0 \\<br />
0 & 0 & 4 & 3 \\<br />
\end{pmatrix}.</math><br />
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==Usage==<br />
One variant of the [[QR algorithm]] starts with reducing a general matrix into a bidiagonal one,<ref>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 http://www.webcitation.org/5utO4iSnR)</ref><br />
and the [[Singular value decomposition]] uses this method as well.<br />
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==See also==<br />
* [[Diagonal matrix]]<br />
* [[List of matrices]]<br />
* [[LAPACK]]<br />
* [[Bidiagonalization]]<br />
* [[Hessenberg form]] The Hessenberg form is similar, but has more non zero diagonal lines than 2.<br />
* [[Tridiagonal matrix]] with three diagonals<br />
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==References==<br />
* Stewart, G. W. (2001) ''Matrix Algorithms, Volume II: Eigensystems''. Society for Industrial and Applied Mathematics. ISBN 0-89871-503-2.<br />
{{Reflist}}<br />
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==External links==<br />
* [http://www.cs.utexas.edu/users/flame/pubs/flawn53.pdf High performance algorithms] for reduction to condensed (Hessenberg, tridiagonal, bidiagonal) form<br />
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[[Category:Linear algebra]]<br />
[[Category:Sparse matrices]]<br />
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