Abstract: In this chapter we briefly review some facts about the solution of linear systems of equations,Ax=b,(3.1)where A ∈ ℝn × n is square and nonsingular. The linear system (3.1) can be solved using Gaussian elimination with partial pivoting, which is equivalent to factorizing the matrix as a product of triangular matrices.We will also consider overdetermined linear systems, where the matrix A ∈ ℝm × n is rectangular with m > n, and their solution using the least squares method. As we are giving the results only as background, we mostly state them without proofs. For thorough presentations of the theory of matrix decompositions for solving linear systems of equations, see, e.g., [42, 92].Before discussing matrix decompositions, we state the basic result concerning conditions for the existence of a unique solution of (3.1).Proposition 3.1. Let A ∈ ℝn × n and assume that A is nonsingular. Then for any right-hand-side b, the linear system Ax = b has a unique solution.Proof. The result is an immediate consequence of the fact that the column vectors of a nonsingular matrix are linearly independent.3.1 LU DecompositionGaussian elimination can be conveniently described using Gauss transformations, and these transformations are the key elements in the equivalence between Gaussian elimination and LU decomposition. More details on Gauss transformations can be found in any textbook in numerical linear algebra; see, e.g., [42, p. 94].
Publication Year: 2007
Publication Date: 2007-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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