Question: A matrix A is said to have band width k if all entries that are more than k slots away from the main diagonal are

A matrix A is said to have band width k if all entries that are more than k slots away from the main diagonal are zero: aij = 0 whenever |i - j| > k.
(a) Show that a tridiagonal matrix has band width1.
(b) Write down an example of a 6 × 6 matrix of band width 2 and one of band width 3.
(c) Prove that the L and U factors of a regular banded matrix have the same band width.
(d) Find the LU factorization of the matrices you wrote down in part (b).
(e) Use the factorization to solve the system Ax = b, where b is the column vector with all entries equal to 1.
(f) How many arithmetic operations are needed to solve Ax = b if A is banded?
(g) Prove or give a counterexample: the inverse of a banded matrix is banded.

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a If matrix A is tridiagonal then the only nonzero elements in i th row are a ii1 a ii a ii1 So a ij ... View full answer

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