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Unformatted text preview: Lecture 8 Matrix Inverse and LU Decomposition ShangHua 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 nonzero pivots (row exchanges allowed) Inverse, Singular Matrix and Degeneracy Suppose there is a nonzero vector x such that A x = [column vectors of A colinear] 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 GaussJordan Elimination for Computing A1 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 GaussJordan Elimination for Computing A1 3D = =...
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 Spring '09
 BERA
 Algorithms

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