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.

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:

then the quantities k₁ and k₂ are called the lower bandwidth and upper bandwidth, respectively. 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 ${\displaystyle |i-j|>k}$.

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 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

is stored as the 6-by-3 matrix