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:

<math>\begin{pmatrix}

1 & 4 & 0 & 0 \\ 0 & 4 & 1 & 0 \\ 0 & 0 & 3 & 4 \\ 0 & 0 & 0 & 3 \\ \end{pmatrix}</math>

and the following matrix is lower bidiagonal:

<math>\begin{pmatrix}

1 & 0 & 0 & 0 \\ 2 & 4 & 0 & 0 \\ 0 & 3 & 3 & 0 \\ 0 & 0 & 4 & 3 \\ \end{pmatrix}.</math>

UsageEdit

One variant of the QR algorithm starts with reducing a general matrix into a bidiagonal one,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }} Accessed: 2010-12-11. (Archived by WebCite at)</ref> and the singular value decomposition (SVD) uses this method as well.