Lecture Notes 5 - Untitled Lecture 05 Outline 1 The Matrix...

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Untitled 1 September 18, 2007 Outline: 1) The Matrix Inverse: A -1 (cont'd) The Mechanics: Gauss-Jordan Elim (3x3 example) The Theory: Proof: invertibility iff n pivots 2) The LU factorization: A=LU (the right way to solve Ax =b ) The idea of factorizations Finding L=E -1 Examples of A=LU Understanding the LU factorization Using A=LU to solve Ax =b Operation Costs row swaps and PA=LU Matlab Examples Lecture 05: Systems of Linear Equations #4: A -1 and the LU decomposition Inverses of General Square Matrices: Gauss-Jordan Elimination The Big Picture Step 4: Divide by the Pivots Step 3: Eliminate Up (Jordan) Step 2: Eliminate Down (Gauss) Step 1: Form the block matrix [ A I ] Inverses of General Square Matrices: Gauss-Jordan Elimination A 3x3 Example (ugh): A = [ 1 4 3 ; -1 -2 0; 2 2 3] Big Point!: To solve Ax =b use Gaussian Elimination, not x =A -1 b Being Invertible is important... finding A -1 is less important But Elimination is both the mechanism and the proof of invertibility Gauss Elimination is a sufficient test for invertibility!
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