Bidiagonal matrix
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 (SVD) uses this method as well.
Bidiagonalization
Bidiagonalization allows guaranteed accuracy when using floating-point arithmetic to compute singular values.[2]
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)
- ^ 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
- v
- t
- e
- 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
- Adjugate
- Alternating sign
- Augmented
- Bézout
- Carleman
- Cartan
- Circulant
- Cofactor
- Commutation
- Confusion
- Coxeter
- Distance
- Duplication and elimination
- Euclidean distance
- Fundamental (linear differential equation)
- Generator
- Gram
- Hessian
- Householder
- Jacobian
- Moment
- Payoff
- Pick
- Random
- Rotation
- Seifert
- Shear
- Similarity
- Symplectic
- Totally positive
- Transformation
- Cabibbo–Kobayashi–Maskawa
- Density
- Fundamental (computer vision)
- Fuzzy associative
- Gamma
- Gell-Mann
- Hamiltonian
- Irregular
- Overlap
- S
- State transition
- Substitution
- Z (chemistry)
- Mathematics portal
- List of matrices
- Category:Matrices
This article about matrices is a stub. You can help Wikipedia by expanding it. |
- v
- t
- e
This computer-programming-related article is a stub. You can help Wikipedia by expanding it. |
- v
- t
- e