In mathematics, particularly matrix theory, a band matrix or banded matrix is a sparse matrix whose non-zero entries are confined to a diagonal band, comprising the main diagonal and zero or more diagonals on either side.

Band matrix

Bandwidth

Formally, consider an n×n matrix A=(a_i,j ). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constants k₁ and k₂:

${\displaystyle a_{i,j}=0\quad {\mbox{if}}\quad j<i-k_{1}\quad {\mbox{ or }}\quad j>i+k_{2};\quad k_{1},k_{2}\geq 0.\,}$

then the quantities k₁ and k₂ are called the lower bandwidth and upper bandwidth, respectively.^[1] The bandwidth of the matrix is the maximum of k₁ and k₂; in other words, it is the number k such that ${\displaystyle a_{i,j}=0}$ if ${\displaystyle |i-j|>k}$.^[2]

Examples

• A band matrix with k₁ = k₂ = 0 is a diagonal matrix
• A band matrix with k₁ = k₂ = 1 is a tridiagonal matrix
• For k₁ = k₂ = 2 one has a pentadiagonal matrix and so on.
• Triangular matrices
  • For k₁ = 0, k₂ = n−1, one obtains the definition of an upper triangular matrix
  • similarly, for k₁ = n−1, k₂ = 0 one obtains a lower triangular matrix.
• Upper and lower Hessenberg matrices
• Toeplitz matrices when bandwidth is limited.
• Block diagonal matrices
• Shift matrices and shear matrices
• Matrices in Jordan normal form
• A skyline matrix, also called "variable band matrix" – a generalization of band matrix
• The inverses of Lehmer matrices are constant tridiagonal matrices, and are thus band matrices.

Applications

In numerical analysis, matrices from finite element or finite difference problems are often banded. Such matrices can be viewed as descriptions of the coupling between the problem variables; the banded property corresponds to the fact that variables are not coupled over arbitrarily large distances. Such matrices can be further divided – for instance, banded matrices exist where every element in the band is nonzero.

Problems in higher dimensions also lead to banded matrices, in which case the band itself also tends to be sparse. For instance, a partial differential equation on a square domain (using central differences) will yield a matrix with a bandwidth equal to the square root of the matrix dimension, but inside the band only 5 diagonals are nonzero. Unfortunately, applying Gaussian elimination (or equivalently an LU decomposition) to such a matrix results in the band being filled in by many non-zero elements.

Band storage

Band matrices are usually stored by storing the diagonals in the band; the rest is implicitly zero.

For example, a tridiagonal matrix has bandwidth 1. The 6-by-6 matrix

${\displaystyle {\begin{bmatrix}B_{11}&B_{12}&0&\cdots &\cdots &0\\B_{21}&B_{22}&B_{23}&\ddots &\ddots &\vdots \\0&B_{32}&B_{33}&B_{34}&\ddots &\vdots \\\vdots &\ddots &B_{43}&B_{44}&B_{45}&0\\\vdots &\ddots &\ddots &B_{54}&B_{55}&B_{56}\\0&\cdots &\cdots &0&B_{65}&B_{66}\end{bmatrix}}}$

is stored as the 6-by-3 matrix

${\displaystyle {\begin{bmatrix}0&B_{11}&B_{12}\\B_{21}&B_{22}&B_{23}\\B_{32}&B_{33}&B_{34}\\B_{43}&B_{44}&B_{45}\\B_{54}&B_{55}&B_{56}\\B_{65}&B_{66}&0\end{bmatrix}}.}$

A further saving is possible when the matrix is symmetric. For example, consider a symmetric 6-by-6 matrix with an upper bandwidth of 2:

${\displaystyle {\begin{bmatrix}A_{11}&A_{12}&A_{13}&0&\cdots &0\\&A_{22}&A_{23}&A_{24}&\ddots &\vdots \\&&A_{33}&A_{34}&A_{35}&0\\&&&A_{44}&A_{45}&A_{46}\\&sym&&&A_{55}&A_{56}\\&&&&&A_{66}\end{bmatrix}}.}$

This matrix is stored as the 6-by-3 matrix: