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:

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

and the following matrix is lower bidiagonal:

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

The eigenvalues of a bidiagonal matrix (of either type) are given by the entries of the diagonal.

For a square bidiagonal matrix B, the determinant is simply the product of its diagonal elements:

${\displaystyle \det(B)=\prod _{i=1}^{n}b_{ii}}$

Additionally, the inverse of a nonsingular upper bidiagonal matrix B is an upper triangular matrix whose entries can be computed explicitly without solving full systems of linear equations.