Band matrix

From Wikipedia, the free encyclopedia

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

The quantities k₁ and k₂ are the left and right half-bandwidth, respectively. The bandwidth of the matrix is k₁ + k₂ + 1 (in other words, the smallest number of adjacent diagonals to which the non-zero elements are confined).

A band matrix with k₁ = k₂ = 0 is a diagonal matrix; a band matrix with k₁ = k₂ = 1 is a tridiagonal matrix; when k₁ = k₂ = 2 one has a pentadiagonal matrix and so on. If one puts k₁ = 0, k₂ = n−1, one obtains the definition of a lower triangular matrix; similarly, for k₁ = n−1, k₂ = 0 one obtains an upper triangular matrix.

Contents 1 Applications 2 Examples and special cases 3 Notes 4 External links

[edit] 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 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.^{[citation needed]}

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 half-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. As sparse matrices lend themselves to more efficient computation than dense matrices, there has been much research focused on finding ways to minimise the bandwidth (or directly minimise the fill in) by applying permutations to the matrix, or other such equivalence or similarity transformations.^{[citation needed]}

From a computational point of view, working with band matrices is always preferential to working with similarly dimensioned dense 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.

[edit] Examples and special cases

The following are special cases of band matrices:

• Diagonal matrices.
• Tridiagonal matrices.
• Pentadiagonal matrices.
• Upper and lower triangular matrices.
• Upper and lower Hessenberg matrices.
• Block-diagonal matrices.
• Shift matrices and shear matrices.
• Matrices in Jordan normal form.

The inverses of Lehmer matrices are constant tridiagonal matrices, and are thus band matrices.

[edit] Notes

The representation of band matrices in the LAPACK Fortran package is not consistent with that found in the EISPACK package.

[edit] External links

• http://www.netlib.org/lapack/lug/node124.html Information pertaining to LAPACK and band matrices.
• http://www.netlib.org/linalg/html_templates/node89.html#SECTION00930000000000000000 A tutorial on banded matrices and other sparse matrix formats.
• http://www.intel.com/software/products/mkl/docs/mklqref/matrixst.htm An overview of matrix representation.
• http://www.cs.ut.ee/~toomas_l/linalg/lin1/node13.html An overview of band representations.

---