Definition:Band Matrix

Definition

A band matrix is a matrix whose elements outside a band of diagonals around the leading diagonal are all zero.

Examples

Arbitrary Example

This is an example of a band matrix:

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

Tridiagonal Matrix

A tridiagonal matrix is a band matrix in which the elements outside the leading diagonal, the subdiagonal and the superdiagonal are all zero.

$\begin {pmatrix} a & b & 0 & \cdots & 0 & 0 \\ c & d & e & \cdots & 0 & 0 \\ 0 & f & g & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & w & x \\ 0 & 0 & 0 & \cdots & y & z \end {pmatrix}$

Also known as

A band matrix is also known as a banded matrix.

Also see

• Results about band matrices can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): band matrix
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): band matrix
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): banded matrix
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): banded matrix