Solve the system using LU -method. If and then Definition 1.2.1. The transformation defined with the Gauss matrix is called the Gauss transformation . If then for the nonzero pivots of A the Gauss matrices can be found such that is upper triangular. The calculation of M k is possible if .
L is a unit lower triangular matrix.
If the LU factorization exists and A is regular, then the LU factorization is unique and Proof. Suppose k -1 steps have been taken and the matrix has been found. Thus, and Example 1.2.4. By solving the system , i.e., we obtain that By solving the system , i.e., we obtain that Exercise 1.2.1. By using the LU factorization solve the system , where If the principal minors of a rectangular matrix are nonzero, i.e., then A has an LU factorization.