Matrix Bandwidth
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 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.
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