lecture12 - Lecture 12: Gaussian Elimination Daniel Frances...

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Unformatted text preview: Lecture 12: Gaussian Elimination Daniel Frances c 2012 Contents 1 Gaussian Elimination 2 1.1 O n analysis for matrix inversion . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 O n analysis for solving equations . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 The divide by zero problem . . . . . . . . . . . . . . . . . . . . . . . 4 1 1 Gaussian Elimination This is basically the approach covered in introductory linear algebra courses and also the one used in the basic simplex tableau method. The basic principle is that we create an nx2n-dim matrix [ A | I ] and perform row operations on A until we have converted [ A | I ] to [ I | A ]. From basic linear algebra performing row operations on a matrix A is equivalent to pre-multiplying the matrix by another matrix, say A . (For completeness, performing column operations is equivalent to post-multiplying by a matrix). Since the matrix [ A | I ] was converted to [ I | A ] = [ A A | A I ] therefore A = A- 1 , and A- 1 can readily be read off from the right side of the [ I | A- 1 ] matrix. 1.1 O n analysis for matrix inversion With this approach lets see how many operations it takes to invert an nxn-dim matrix with Gaussian elimination. As usual we start with a small example. For now we assume all entriesGaussian elimination....
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lecture12 - Lecture 12: Gaussian Elimination Daniel Frances...

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