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Concept
Band matrix
Summary
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=(ai,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 and , respectively. The of the matrix is the maximum of k1 and k2; in other words, it is the number k such that if .
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.
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.
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.
Official source
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