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:
- [math]\displaystyle{ \begin{pmatrix} 1 & 4 & 0 & 0 \\ 0 & 4 & 1 & 0 \\ 0 & 0 & 3 & 4 \\ 0 & 0 & 0 & 3 \\ \end{pmatrix} }[/math]
and the following matrix is lower bidiagonal:
- [math]\displaystyle{ \begin{pmatrix} 1 & 0 & 0 & 0 \\ 2 & 4 & 0 & 0 \\ 0 & 3 & 3 & 0 \\ 0 & 0 & 4 & 3 \\ \end{pmatrix}. }[/math]
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 https://www.webcitation.org/5utO4iSnR)
- ↑ 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.
External links
- High performance algorithms for reduction to condensed (Hessenberg, tridiagonal, bidiagonal) form
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