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: Band form of sparse matrices 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. 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. The Cuthill–McKee algorithm can be used to reduce the bandwidth of a sparse symmetric matrix. There are, however, matrices for which the reverse 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 is NP-hard. See also Diagonal matrix Graph bandwidth Notes References . . . . . External links Information pertaining to LAPACK and band matrices A tutorial on banded matrices and other sparse matrix formats Sparse matrices