Definition:Bidiagonal Matrix
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Definition
A bidiagonal matrix is a square matrix in which the elements outside the leading diagonal and either the subdiagonal or the superdiagonal are all zero.
Upper
An upper bidiagonal matrix is a square matrix in which the elements outside the leading diagonal and superdiagonal are all zero.
- $\begin {pmatrix} a & b & 0 & \cdots & 0 & 0 \\ 0 & c & d & \cdots & 0 & 0 \\ 0 & 0 & e & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & x & y \\ 0 & 0 & 0 & \cdots & 0 & z \end {pmatrix}$
That is, a square matrix such that $a_{i j} = 0$ when $i > j$ or $j > i + 1$.
Lower
A lower bidiagonal matrix is a square matrix in which the elements outside the leading diagonal and subdiagonal are all zero.
- $\begin {pmatrix} a & 0 & 0 & \cdots & 0 & 0 \\ b & c & 0 & \cdots & 0 & 0 \\ 0 & d & e & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & x & 0 \\ 0 & 0 & 0 & \cdots & y & z \end {pmatrix}$
That is, a square matrix such that $a_{i j} = 0$ when $i < j$ or $i > j + 1$.
Also see
- Results about bidiagonal matrices can be found here.
Sources
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): bidiagonal matrix