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). 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
