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lecture8 - Lecture 8 Matrix Inverse and LU Decomposition...

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Unformatted text preview: Lecture 8 Matrix Inverse and LU Decomposition Shang-Hua Teng Inverse Matrices In high dimensions I AA A A A b A x b Ax = = = =---- 1 1 1 1 such that? matrix a there Is write? Can we Uniqueness of Inverse Matrices ( 29 ( 29 C IC C BA BAC AC B BI B C B I AC I BA = = = = = = = = = : Proof then and Inverse and Linear System b A x b A Ix b A Ax A b A b Ax A 1 1 1 1 1 : Proof by given solution unique a has then invertible is if----- = = = = Inverse and Linear System Therefore, the inverse of A exists if and only if elimination produces n non-zero pivots (row exchanges allowed) Inverse, Singular Matrix and Degeneracy Suppose there is a nonzero vector x such that A x = [column vectors of A co-linear] then A cannot have an inverse : Proof 1 1 = =-- x A Ax A Contradiction: So if A is invertible, then A x =0 can only have the zero solution x = One More Property Proof ( 29 1 1 1--- = A B AB ( 29 ( 29 I B B B A A B AB A B = = =----- 1 1 1 1 1 So ( 29 1 1 1 1---- = A B C ABC Gauss-Jordan Elimination for Computing A-1 1D 1 implies 1- = = a x ax 2D = = = 1 1 then 1 and 1 2 2 1 1 22 21 12 11 2 1 22 21 12 11 2 1 22 21 12 11 y x y x a a a a y y a a a a x x a a a a Gauss-Jordan Elimination for Computing A-1 3D = =...
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lecture8 - Lecture 8 Matrix Inverse and LU Decomposition...

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