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7 Gaussian Elimination and LU Factorization - IIT

7 Gaussian Elimination and LU FactorizationIn this final section on matrix Factorization methods for solvingAx=bwe want totake a closer look at Gaussian Elimination (probably the best known method for solvingsystems of linear equations).The basic idea is to use left-multiplication ofA Cm mby (elementary) lowertriangular matrices ,L1, L2, .. , Lm 1to convertAto upper triangular form, ,Lm 1Lm 2.. L2L1 =eLA= that the product of lower triangular matrices is a lower triangular matrix, andthe inverse of a lower triangular matrix is also lower triangular. Therefore, LA=U A=LU,whereL= L 1. This approach can be viewed astriangular Why Would We Want to Do This?

systems of linear equations). The basic idea is to use left-multiplication of A ∈Cm×m by (elementary) lower triangular matrices, L 1,L 2, ...

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