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:

$\left(\begin{array}{cccc}1& 4& 0& 0\\ 0& 4& 1& 0\\ 0& 0& 3& 4\\ 0& 0& 0& 3\end{array}\right)$

and the following matrix is lower bidiagonal:

$\left(\begin{array}{cccc}1& 0& 0& 0\\ 2& 4& 0& 0\\ 0& 3& 3& 0\\ 0& 0& 4& 3\end{array}\right).$

The eigenvalues of a bidiagonal matrix (of either type) are given by the entries of the diagonal.

Usage

One variant of the QR algorithm starts with reducing a general matrix into a bidiagonal one, 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.

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.

External links

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

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