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-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:

{\begin{pmatrix}1&4&0&0\\0&4&1&0\\0&0&3&4\\0&0&0&3\\\end{pmatrix}}

and the following matrix is lower bidiagonal:

{\begin{pmatrix}1&0&0&0\\2&4&0&0\\0&3&3&0\\0&0&4&3\\\end{pmatrix}}.

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

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)

^[1]

External links

High performance algorithms for reduction to condensed (Hessenberg, tridiagonal, bidiagonal) form

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^ Fernando, K.V. (1 April 2007). "Computation of exact inertia and inclusions of eigenvalues (singular values) of tridiagonal (bidiagonal) matrices". Linear Algebra and its Applications. 422 (1): 77–99. doi:10.1016/j.laa.2006.09.008. Retrieved 6 July 2022.

[1] 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)

[2] Fernando, K.V. (1 April 2007). "Computation of exact inertia and inclusions of eigenvalues (singular values) of tridiagonal (bidiagonal) matrices". Linear Algebra and its Applications. 422 (1): 77–99. doi:10.1016/j.laa.2006.09.008. Retrieved 6 July 2022.

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 * Stewart, G. W. (2001) ''Matrix Algorithms, Volume II: Eigensystems''. Society for Industrial and Applied Mathematics. {{ISBN|0-89871-503-2}}.
 {{Reflist}}
+<ref>{{cite journal |last1=Fernando |first1=K.V. |title=Computation of exact inertia and inclusions of eigenvalues (singular values) of tridiagonal (bidiagonal) matrices |journal=Linear Algebra and its Applications |date=1 April 2007 |volume=422 |issue=1 |pages=77-99 |doi=10.1016/j.laa.2006.09.008 |url=https://www.sciencedirect.com/science/article/pii/S0024379506004228 |access-date=6 July 2022 |ref=inertia}}</ref>
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Revision as of 22:01, 6 July 2022

Usage

Bidiagonalization

See also

References

External links