Bidiagonal matrix

Summary

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 edit

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 edit

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

See also edit

References edit

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

External links edit

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