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 (SVD) uses this method as well.

Bidiagonalization

Bidiagonalization allows guaranteed accuracy when using floating-point arithmetic to compute singular values.^[2]

References

Stewart, G.W. (2001). Eigensystems. Matrix Algorithms. Vol. 2. Society for Industrial and Applied Mathematics. ISBN 0-89871-503-2.

^ Anatolyevich, Bochkanov Sergey (2010-12-11). "Matrix operations and decompositions — Other operations on general matrices — SVD decomposition". ALGLIB User Guide, ALGLIB Project. Accessed: 2010-12-11. (Archived by WebCite at)
^ 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. S2CID 122729700.

External links

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

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[1] Anatolyevich, Bochkanov Sergey (2010-12-11). "Matrix operations and decompositions — Other operations on general matrices — SVD decomposition". ALGLIB User Guide, ALGLIB Project. Accessed: 2010-12-11. (Archived by WebCite at)

[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. S2CID 122729700.

[1]

[2]

@@ Line 1: / Line 1: @@
-A '''bidiagonal matrix''' is a matrix with non-zero entries along the main diagonal and ''either'' the diagonal above or the diagonal below.
+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.
-So that means there are two non zero diagonal 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''':
@@ Line 21: / Line 18: @@
 & 3 & 3 & 0 \\
 & 0 & 4 & 3 \\
-\end{pmatrix}</math>
+\end{pmatrix}.</math>
 ==Usage==
+One variant of the [[QR algorithm]] starts with reducing a general matrix into a bidiagonal one,<ref>{{cite web |first=Bochkanov Sergey |last=Anatolyevich |title=Matrix operations and decompositions — Other operations on general matrices — SVD decomposition |date=2010-12-11 |work=ALGLIB User Guide, ALGLIB Project |url=https://www.alglib.net/matrixops/general/svd.php}} Accessed: 2010-12-11. (Archived by WebCite at)</ref>
+and the [[singular value decomposition]] (SVD) uses this method as well.
+===Bidiagonalization===
-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>
+{{Main|Bidiagonalization}}
-and the [[Singular value decomposition]] uses this method as well.
+Bidiagonalization allows guaranteed accuracy when using [[floating-point arithmetic]] to compute singular values.<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 |s2cid=122729700 |ref=inertia|doi-access=free }}</ref>
-==See also==
+{{expand Section|date=January 2017}}
+==See also==
-* [[Diagonal matrix]]
 * [[List of matrices]]
 * [[LAPACK]]
+* [[Hessenberg form]] — The Hessenberg form is similar, but has more non-zero diagonal lines than 2.
-* [[Bidiagonalization]]
-* [[Hessenberg form]] The Hessenberg form is similar, but has more non zero diagonal lines than 2.
-* [[Tridiagonal matrix]] with three diagonals
 ==References==
+{{refbegin}}
-* Stewart, G. W. (2001) ''Matrix Algorithms, Volume II: Eigensystems''. Society for Industrial and Applied Mathematics. ISBN 0-89871-503-2.
+* {{cite book |first=G.W. |last=Stewart |title=Eigensystems |series=Matrix Algorithms |volume=2 |publisher=Society for Industrial and Applied Mathematics |location= |date=2001 |isbn=0-89871-503-2 }}
+{{refend}}
 {{Reflist}}
 ==External links==
 * [http://www.cs.utexas.edu/users/flame/pubs/flawn53.pdf High performance algorithms] for reduction to condensed (Hessenberg, tridiagonal, bidiagonal) form
+{{Matrix classes}}
 [[Category:Linear algebra]]
 [[Category:Sparse matrices]]
-{{algebra-stub}}
-{{compu-prog-stub}}
+{{matrix-stub}}
-[[sl:Bidiagonalna matrika]]
+{{compu-prog-stub}}
-[[sv:Bidiagonal matris]]
-[[es: Matriz bidiagonal]]

v t e Matrix classes
Explicitly constrained entries	Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Bisymmetric Block-diagonal Block Block tridiagonal Boolean Cauchy Centrosymmetric Conference Complex Hadamard Copositive Diagonally dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer Logical Matrix unit Metzler Moore Nonnegative Pentadiagonal Permutation Persymmetric Polynomial Quaternionic Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Vandermonde Walsh Z
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