Systems with Band Matrices

In many applications the matrix A of the system of equations is a band matrix, i.e., the unknown quantity appears with a nonzero coefficient only in the i-th equation and some ``neighbouring'' to it equations.

Proposition 6.5.1. Let A=LU be the LU factorization of the band matrix . If the upper band width of the matrix A is q and the lower band width is p, then the matrix U has the upper band width q and the matrix L has the lower band width p.

Proof. We will prove it by induction. In the case n=1, this assertion is valid. Let us show the admissibility of the step of induction. Let the proposition be correct for an matrix A. Let the matrix A be given in the form

The following equality is valid:

Since in the vectors and at the most only the first p and q coordinates are different from zero, then the matrix has the upper band width p and the lower band width q. The matrix is a matrix, and, hence where U₁ has the upper band width q and L₁ has the lower band width p. The matrices

and

have the band width p and q, respectively, and

Problem 6.5.1. Find for the LU factorization of the matrix given in example 6.2.1 the upper and lower band width for the matrices and U.