ABSTRACT
In this chapter, our main objective is to prove the general decomposition of any matrix into the product of a unit lower triangular matrix by an upper triangular one. Such decompositions formalize the procedures for solving systems of linear equations by Gauss elimination. This material is at the heart of matrix computations and constitutes the basis for computing determinant and inverse of square matrices. More specific details can be found in books such as Ciarlet [19], Poole [56], Trefethen [69], and Golub and Van Loan [33]. Gauss reduction is the main procedure for obtaining an LU decomposition or PLU decompositions. In the first case, the reduction is said to be a Naive Gauss Reduction (Section 3.3) while in the second case (Section 3.4), the reduction is referred to as Gauss Reduction with Partial Pivoting, scaled or unscaled. Unless stated otherwise, all matrices are considered to be dense, real and square matrices.
