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:
:$\begin 1 & 4 & 0 & 0 \\ 0 & 4 & 1 & 0 \\ 0 & 0 & 3 & 4 \\ 0 & 0 & 0 & 3 \\ \end$

and the following matrix is lower bidiagonal:

:$\begin 1 & 0 & 0 & 0 \\ 2 & 4 & 0 & 0 \\ 0 & 3 & 3 & 0 \\ 0 & 0 & 4 & 3 \\ \end.$

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) ''Matrix Algorithms, Volume II: Eigensystems''. Society for Industrial and Applied Mathematics. .

External links

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

Linear algebra
Sparse matrices

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