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Unformatted text preview: The Householder transformation in numerical linear algebra John Kerl February 3, 2008 Abstract In this paper I define the Householder transformation, then put it to work in several ways: To illustrate the usefulness of geometry to elegantly derive and prove seemingly algebraic properties of the transform; To demonstrate an application to numerical linear algebra specifically, for matrix determinants and inverses; To show how geometric notions of determinant and matrix norm can be used to easily understand round-off error in Householder and Gaussian-elimination methods. These are notes to accompany a talk given to graduate students in mathematics. However, most of the content (except references to the orthogonal group) should be accessible to an undergraduate with a course in introductory linear algebra. 1 Contents Contents 2 1 Linear algebra 3 1.1 Geometric meanings of determinant and matrix norm . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Computation of determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Computation of matrix inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Error propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Gaussian elimination 8 2.1 Row reduction using Gaussian elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Gaussian elimination without pivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Gaussian elimination with pivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Householder transformations 11 3.1 Geometric construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Construction with specified source and destination . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Properties of Q , obtained algebraically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Properties of Q , obtained geometrically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.5 Repeated Householders for upper-triangularization . . . . . . . . . . . . . . . . . . . . . . . . 14 3.6 Householders for column-zeroing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.7 Computation of determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.8 Computation of matrix inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.9 Rotation matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.10 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4 Acknowledgments 18 References 19 Index 20 2 1 Linear algebra In this paper I compare and contrast two techniques for computation of determinants and inverses of square matrices: the more-familiar Gaussian-elimination method, and the less-familiar Householder method. Imatrices: the more-familiar Gaussian-elimination method, and the less-familiar Householder method....
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This note was uploaded on 03/27/2008 for the course MATH 27 taught by Professor Jung during the Spring '08 term at Texas A&M.

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dhh - The Householder transformation in numerical linear...

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