A banded matrix is a matrix whose only nonzero elements are located in a band ar

Question

A banded matrix is a matrix whose only nonzero elements are located in a band around the main diagonal. For example, if A = [a_1 d_1 b_1 a_2 c_1 b_2 c_2 d_n-1 c_n-2 b_n-1 a_n] then the number of nonzero diagonals above the main diagonal is two (the b_1's and c_i's) and the number of nonzero diagonals below the main diagonal is one (the d_i's). Let A be a banded matrix of size n times n with three diagonals above the main diagonal and four diagonals below it. Let A = L + U be the LU decomposition of A. What can be said about the structure of L and U, i.e., where are the nonzero elements of L and U?

Explanation / Answer

A will be factored into lower triangular matrix L, and upper triangular matrix U.

All non-zero elements of L will be 1 and they will be four diagonals below the main diagonal.

All non-zero elements of U will be 2 and they will be three diagonals below the main diagonal.

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