Band Matrix - Matrix Bandwidth

Matrix Bandwidth

Formally, consider an n×n matrix A=(a_i,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 left and right half-bandwidth, respectively (Golub & Van Loan 1996, §1.2.1). The bandwidth of the matrix is k₁ + k₂ + 1 (in other words, it is the smallest number of adjacent diagonals to which the non-zero elements are confined).

A matrix is called a band matrix or banded matrix if its bandwidth is reasonably small.

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 an upper triangular matrix; similarly, for k₁ = n−1, k₂ = 0 one obtains a lower triangular matrix.