This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 = =...
View
Full
Document
This note was uploaded on 03/08/2010 for the course CS 232 taught by Professor Bera during the Spring '09 term at BU.
 Spring '09
 BERA
 Algorithms

Click to edit the document details