Linear and multilinear algebra; matrix theory Definition

Band matrix

Definition

Let $R$ be a ring, let $A = (a_{ij})$ be an $m \times n$ matrix over $R$ , and let $p, q \ge 0$ be integers. The matrix $A$ is a band matrix with lower bandwidth $p$ and upper bandwidth $q$ when $a_{ij} = 0 \quad \text{whenever } i > j + p \text{ or } j > i + q.$ The bandwidth of $A$ is often taken as the least $k$ such that the condition holds with $p = q = k$ .

Remarks

Example: $\begin{pmatrix} 1 & 2 & 0 & 0 \\ 3 & 4 & 5 & 0 \\ 6 & 7 & 8 & 9 \\ 0 & 1 & 2 & 3 \end{pmatrix}$ has lower bandwidth $2$ and upper bandwidth $1$ : the entries with $i > j + 2$ or $j > i + 1$ are all $0$ . Non-example: $\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ has no upper bandwidth $q = 1$ , because $a_{13} = 1$ and $3 > 1 + 1$ . With $p = q = 0$ one gets the diagonal matrices, and with $p = q = 1$ the tridiagonal matrices. A square matrix with $p = 0$ is upper triangular. The term is mostly used when $p$ and $q$ are small compared with the size of the matrix.

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Matrix concept
Ring concept
Band matrix definition

Tag	`T12HYJTXJF`
URI	`m2://T12HYJTXJF`
Kind	definition
Writing	W0 stub
MSC	15B99, 65F05
Defines	Band matrix
Uses	Ring, Matrix
Changed	2 November 2026

Band matrix

Definition

Remarks

Reading path

Further reading

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