An LU factorization of a matrix involves writing the given matrix as the product of a lower triangular matrix which has the main diagonal consisting entirely of ones, L, and an upper triangular matrix U in the indicated order. The L goes with “lower” and the U with “upper”. It turns out many matrices can be written in this way and when this is possible, people get excited about slick ways of solving the system of equations, Ax = y. The method lacks generality but is of interest just the same.
Example 5.1.1 Can you write
To do so you would need
Therefore, b = 1 and a = 0. Also, from the bottom rows, xa = 1 which can’t happen and have a = 0. Therefore, you can’t write this matrix in the form LU. It has no LU factorization. This is what I mean above by saying the method lacks generality.
Which matrices have an LU factorization? It turns out it is those whose row reduced echelon form can be achieved without switching rows and which only involve row operations of type 3 in which row j is replaced with a multiple of row i added to row j for i < j.