Bidiagonal matrix

In linear algebra , a bidiagonal matrix is a square matrix that only contains entries other than zero in the main diagonal and in one of the first two secondary diagonals . There are lower and upper bidiagonal matrices, the terms are to be understood analogously to such a designation of triangular matrices . ${\displaystyle A}$

Accordingly, an upper diagonal matrix always has the form ${\displaystyle n\times n}$ ${\displaystyle A}$

${\displaystyle A={\begin{pmatrix}a_{1,1}&a_{1,2}&0&\dots &0\\0&a_{2,2}&a_{2,3}&\ddots &\vdots \\0&\ddots &\ddots &\ddots &0\\\vdots &\ddots &\ddots &\ddots &a_{n-1,n}\\0&\dots &0&0&a_{n,n}\end{pmatrix}}}$.

Bidiagonal matrices are a special case of tridiagonal matrices , which in turn represent a special case of both ribbon matrices and Hessenberg matrices .

use

Bidiagonal matrices occur z. B. in the following situations:

• as Jordan blocks in the Jordan normal form ,
• as an intermediate step in the calculation of the singular value decomposition .

See also

• Tridiagonal matrix
• Band matrix
• Hessenberg matrix

literature

1. ↑ Wolfgang Dahmen: Numerics for Engineers and Natural Scientists , p. 149.