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Unformatted text preview: Lecture 17 Introduction to Eigenvalue Problems ShangHua Teng 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 websearch • It is the key ingredient for the analysis of Markov process, sampling algorithms, and various approximation algorithms in computer science 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 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 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 Examples: Eigenvalues and Eigenvectors = = = = = 1 , 2 , 1 , 1 2 1 2 2 1 1 x x A λ λ Note: x 1 and x 2 are perpendicular to each other Examples: Eigenvalues and Eigenvectors = = = = = 1 1 , 2 , 1 , 1 2 1 1 2 2 1 1 x x A λ λ Note: x 1 and x 2 are not perpendicular to each other Examples: Eigenvalues and Eigenvectors  = =...
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This note was uploaded on 03/08/2010 for the course CS 232 taught by Professor Bera during the Spring '09 term at BU.
 Spring '09
 BERA
 Algorithms

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