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lecture17.0 - Lecture 17 Introduction to Eigenvalue...

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Lecture 17 Introduction to Eigenvalue Problems Shang-Hua Teng
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Eigenvalue Problems Eigenvalue problems occur in many areas of science and engineering E.g., Structure analysis It is important for analyzing numerical and linear algebra algorithms Impact of roundoff errors and precision requirement It is widely used in information management and web-search It is the key ingredient for the analysis of Markov process, sampling algorithms, and various approximation algorithms in computer science
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Eigenvalues and Eigenvectors Standard Eigenvalue Problem: Given an n by n matrix A, find a scalar λ and nonzero vector x such that A x = λ x 2200 λ is eigenvalue, and x is corresponding eigenvector
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Spectrum of Matrices Spectrum(A) = λ (A) = set of all eigenvalues of A Spectral radius (A) = ρ (A) = max {| λ |: λ in λ (A)} Spectral analysis Spectral methods
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Geometric Interpretation Matrix expands or shrinks any vector lying in direction of eigenvector by scalar factor Expansion of contraction factor given by corresponding eigenvalue λ Eigenvalues and eigenvectors decompose complicated behavior of general linear transformation into simpler actions
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Examples: Eigenvalues and Eigenvectors = = = = = 1 0 , 2 , 0 1 , 1 2 0 0 1 2 2 1 1 x x A λ λ Note: x 1 and x 2 are perpendicular to each other
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Examples: Eigenvalues and Eigenvectors = = = = = 1 1 , 2 , 0 1 , 1 2 0 1 1 2 2 1 1 x x A λ λ Note: x 1 and x 2 are not perpendicular to each other
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Examples: Eigenvalues and Eigenvectors - = = = = - - = 1 1 , 4 , 1 1 , 2 3 1 1 3 2 2 1 1 x x A λ λ Note: x 1 and x 2 are perpendicular to each other
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