Band matrix

In mathematics, particularly matrix theory, a band 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 ${\displaystyle |i-j|>k}$.

A matrix is called a band matrix or banded matrix if its bandwidth is reasonably small.

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 bandedness 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. These often arise when discretising one-dimensional problems.

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