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Bidiagonal matrix
In mathematics, a bidiagonal matrix is a 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 uses this method as well.
See also
Diagonal matrix
List of matrices
LAPACK
Bidiagonalization
Hessenberg form The Hessenberg form is similar, but has more non zero diagonal lines than 2.
Tridiagonal matrix with three diagonals
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 http://www.webcitation.org/5utO4iSnR)
External links
High performance algorithms for reduction to condensed (Hessenberg, tridiagonal, bidiagonal) form
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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