Sparse Matrix - Band Matrix

Band Matrix

An important special type of sparse matrices is band matrix, defined as follows. The lower bandwidth of a matrix A is the smallest number p such that the entry a_ij vanishes whenever i > j + p. Similarly, the upper bandwidth is the smallest p such that a_ij = 0 whenever i < j − p (Golub & Van Loan 1996, §1.2.1). For example, a tridiagonal matrix has lower bandwidth 1 and upper bandwidth 1. As another example, the following sparse matrix has lower and upper bandwidth both equal to 3. Notice that zeros are represented with dots.

Matrices with reasonably small upper and lower bandwidth are known as band matrices and often lend themselves to simpler algorithms than general sparse matrices; or one can sometimes apply dense matrix algorithms and gain efficiency simply by looping over a reduced number of indices.

By rearranging the rows and columns of a matrix A it may be possible to obtain a matrix A’ with a lower bandwidth. A number of algorithms are designed for bandwidth minimization.