# chap04

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Eigenvalue Problems Existence, Uniqueness, and Conditioning Computing Eigenvalues and Eigenvectors Scientific Computing: An Introductory Survey Chapter 4 – Eigenvalue Problems Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted for noncommercial, educational use only. Michael T. Heath Scientific Computing 1 / 87

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Eigenvalue Problems Existence, Uniqueness, and Conditioning Computing Eigenvalues and Eigenvectors Outline 1 Eigenvalue Problems 2 Existence, Uniqueness, and Conditioning 3 Computing Eigenvalues and Eigenvectors Michael T. Heath Scientific Computing 2 / 87
Eigenvalue Problems Existence, Uniqueness, and Conditioning Computing Eigenvalues and Eigenvectors Eigenvalue Problems Eigenvalues and Eigenvectors Geometric Interpretation Eigenvalue Problems Eigenvalue problems occur in many areas of science and engineering, such as structural analysis Eigenvalues are also important in analyzing numerical methods Theory and algorithms apply to complex matrices as well as real matrices With complex matrices, we use conjugate transpose, A H , instead of usual transpose, A T Michael T. Heath Scientific Computing 3 / 87

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Eigenvalue Problems Existence, Uniqueness, and Conditioning Computing Eigenvalues and Eigenvectors Eigenvalue Problems Eigenvalues and Eigenvectors Geometric Interpretation Eigenvalues and Eigenvectors Standard eigenvalue problem : Given n × n matrix A , find scalar λ and nonzero vector x such that A x = λ x λ is eigenvalue , and x is corresponding eigenvector λ may be complex even if A is real Spectrum = λ ( A ) = set of eigenvalues of A Spectral radius = ρ ( A ) = max {| λ | : λ λ ( A ) } Michael T. Heath Scientific Computing 4 / 87
Eigenvalue Problems Existence, Uniqueness, and Conditioning Computing Eigenvalues and Eigenvectors Eigenvalue Problems Eigenvalues and Eigenvectors Geometric Interpretation Geometric Interpretation Matrix expands or shrinks any vector lying in direction of eigenvector by scalar factor Expansion or contraction factor is given by corresponding eigenvalue λ Eigenvalues and eigenvectors decompose complicated behavior of general linear transformation into simpler actions Michael T. Heath Scientific Computing 5 / 87

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Eigenvalue Problems Existence, Uniqueness, and Conditioning Computing Eigenvalues and Eigenvectors Eigenvalue Problems Eigenvalues and Eigenvectors Geometric Interpretation Examples: Eigenvalues and Eigenvectors A = 1 0 0 2 : λ 1 = 1 , x 1 = 1 0 , λ 2 = 2 , x 2 = 0 1 A = 1 1 0 2 : λ 1 = 1 , x 1 = 1 0 , λ 2 = 2 , x 2 = 1 1 A = 3 - 1 - 1 3 : λ 1 = 2 , x 1 = 1 1 , λ 2 = 4 , x 2 = 1 - 1 A = 1 . 5 0 . 5 0 . 5 1 . 5 : λ 1 = 2 , x 1 = 1 1 , λ 2 = 1 , x 2 = - 1 1 A = 0 1 - 1 0 : λ 1 = i, x 1 = 1 i , λ 2 = - i, x 2 = i 1 where i = - 1 Michael T. Heath Scientific Computing 6 / 87
Eigenvalue Problems Existence, Uniqueness, and Conditioning Computing Eigenvalues and Eigenvectors Characteristic Polynomial Relevant Properties of Matrices Conditioning Characteristic Polynomial Equation Ax = λ x is equivalent to ( A - λ I ) x = 0 which has nonzero solution x if, and only if, its matrix is singular Eigenvalues of A are roots λ i of

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