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=(ai,j ). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constants k1 and k2: 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. These often arise when discretising one-dimensional problems.Band matrices are usually stored by storing the diagonals in the band; the rest is implicitly zero.From a computational point of view, working with band matrices is always preferential to working with similarly dimensioned square matrices. A band matrix can be likened in complexity to a rectangular matrix whose row dimension is equal to the bandwidth of the band matrix. Thus the work involved in performing operations such as multiplication falls significantly, often leading to huge savings in terms of calculation time and complexity.