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:
and the following matrix is lower bidiagonal:
The eigenvalues of a bidiagonal matrix 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 uses this method as well.