Inmathematics, particularlymatrix theory, aband matrix orbanded matrix is asparse matrix whose non-zero entries are confined to a diagonalband, comprising themain diagonal and zero or more diagonals on either side.

Formally, consider ann×n matrixA=(a_i,j). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constantsk₁ andk₂:

then the quantitiesk₁ andk₂ are called thelower bandwidth andupper bandwidth, respectively.^[1] Thebandwidth of the matrix is the maximum ofk₁ andk₂; in other words, it is the numberk such that${\displaystyle a_{i,j}=0}$ if${\displaystyle |i-j|>k}$.^[2]

• A band matrix withk₁ =k₂ = 0 is adiagonal matrix, with bandwidth 0.
• A band matrix withk₁ =k₂ = 1 is atridiagonal matrix, with bandwidth 1.
• Fork₁ =k₂ = 2 one has a pentadiagonal matrix and so on.
• Triangular matrices
  • Fork₁ = 0,k₂ =n−1, one obtains the definition of an uppertriangular matrix
  • similarly, fork₁ =n−1,k₂ = 0 one obtains a lower triangular matrix.
• Upper and lowerHessenberg matrices
• Toeplitz matrices when bandwidth is limited.
• Block diagonal matrices
• Shift matrices andshear matrices
• Matrices inJordan normal form
• Askyline matrix, also called "variable band matrix" – a generalization of band matrix
• The inverses ofLehmer matrices are constant tridiagonal matrices, and are thus band matrices.

Innumerical analysis, matrices fromfinite element orfinite 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 thesquare root of the matrix dimension, but inside the band only 5 diagonals are nonzero. Unfortunately, applyingGaussian elimination (or equivalently anLU 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, atridiagonal matrix has bandwidth 1. The 6-by-6 matrix

is stored as the 6-by-3 matrix

A further saving is possible when the matrix is symmetric. For example, consider a symmetric 6-by-6 matrix with an upper bandwidth of 2:

This matrix is stored as the 6-by-3 matrix:

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

As sparse matrices lend themselves to more efficient computation than dense matrices, as well as in more efficient utilization of computer storage, 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.^[3]

TheCuthill–McKee algorithm can be used to reduce the bandwidth of a sparsesymmetric matrix. There are, however, matrices for which thereverse Cuthill–McKee algorithm performs better. There are many other methods in use.

The problem of finding a representation of a matrix with minimal bandwidth by means of permutations of rows and columns isNP-hard.^[4]

• Diagonal matrix
• Graph bandwidth
• Random matrix

• Atkinson, Kendall E. (1989),An Introduction to Numerical Analysis, John Wiley & Sons,ISBN 0-471-62489-6.
• Davis, Timothy A. (2006),Direct Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics,ISBN 978-0-898716-13-9.
• Feige, Uriel (2000), "Coping with the NP-Hardness of the Graph Bandwidth Problem",Algorithm Theory - SWAT 2000, Lecture Notes in Computer Science, vol. 1851, pp. 129–145,doi:10.1007/3-540-44985-X_2.
• Golub, Gene H.;Van Loan, Charles F. (1996),Matrix Computations (3rd ed.), Baltimore: Johns Hopkins,ISBN 978-0-8018-5414-9.
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007),"Section 2.4",Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press,ISBN 978-0-521-88068-8, archived fromthe original on 2016-03-04, retrieved2011-08-08.

• Information pertaining to LAPACK and band matrices
• A tutorial on banded matrices and other sparse matrix formats

• Sparse matrices

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• Short description matches Wikidata