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:

${\begin{pmatrix}1&4&0&0\\0&4&1&0\\0&0&3&4\\0&0&0&3\\\end{pmatrix}}$

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, and the Singular value decomposition uses this method as well.

Bidiagonalization

Main article: Bidiagonalization

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.

External links

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