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 (SVD) uses this method as well.

Bidiagonalization
Bidiagonalization allows guaranteed accuracy when using floating-point arithmetic to compute singular values.