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:

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

and the following matrix is lower bidiagonal:

${\displaystyle {\begin{pmatrix}1&0&0&0\\2&4&0&0\\0&3&3&0\\0&0&4&3\\\end{pmatrix)).}$

Usage

One variant of the QR algorithm starts with reducing a general matrix into a bidiagonal one,^[1] and the singular value decomposition (SVD) uses this method as well.

Bidiagonalization

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