lecture17

lecture17 - Lecture 17 Introduction to Eigenvalue Problems...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lecture 17 Introduction to Eigenvalue Problems Shang-Hua 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 web-search • 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 - = =...
View Full Document

This note was uploaded on 03/08/2010 for the course CS 232 taught by Professor Bera during the Spring '09 term at BU.

Page1 / 30

lecture17 - Lecture 17 Introduction to Eigenvalue Problems...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online