I am trying to decompose solving a system of linear equations using block decomposition where I have an (n x n) matrix A, which is a lower/upper triangular matrix, and a matrix B, which is a rectangular matrix (n x m). The matrices are dense and non-zero.
I am trying to solve LZ = B using the lower triangular matrix, but want to do the work on a series of blocks, such that I decompose B into (n x n) blocks and solve them independently. For example: do A Z1 = B1, A Z2 = B2, ... A Zn = Bn.
Pictured here: https://docs.google.com/drawings/d/12yvLvcVZYgsJx2s4d2lbht9nBly-LA75v6bLtOY9uKQ/edit
Can I do each of these independently or are there dependencies between the various Z and B blocks?
Also could I do the same method, but using the upper triangular matrix of A and solve for WU = C, where C is an (m x n) matrix?