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(a) Suppose that the squared matrix A is symmetric and all its eigenvalues are different. Show that all the eigenvectors of the matrix A are linearly independent. e

(a) Suppose that the squared matrix A is symmetric and all its eigenvalues are different. Show that all the eigenvectors of the matrix A are linearly independent. e

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.5: Iterative Methods For Computing Eigenvalues
Problem 3EQ
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(PageRank): (a) Suppose that the squared matrix A is symmetric and all its eigenvalues are different. Show that all the eigenvectors of the matrix A are linearly independent. - (b) Consider the following WWW (directed) graph. Let w=[w(1) w(2) w(3)] " You need to formulate the PageRank problem as Mw=w, where W21:W22:W23=3:1:2 and w32:w33=1:2. (c) Show that the power method for solving Mw=w in Question 1b can converge to the correct answer w (i.e., the eigenvector of M with the eigenvalue = 1). - [Hint: Use the fact (without proof) that all the eigenvalues of this matrix M are different.] [Remark: In this question, you must not state that it converges by using the power method.] e (d) Use the power method to solve Mw=wwith ɛ=0.1. e
Transcribed Image Text:(PageRank): (a) Suppose that the squared matrix A is symmetric and all its eigenvalues are different. Show that all the eigenvectors of the matrix A are linearly independent. - (b) Consider the following WWW (directed) graph. Let w=[w(1) w(2) w(3)] " You need to formulate the PageRank problem as Mw=w, where W21:W22:W23=3:1:2 and w32:w33=1:2. (c) Show that the power method for solving Mw=w in Question 1b can converge to the correct answer w (i.e., the eigenvector of M with the eigenvalue = 1). - [Hint: Use the fact (without proof) that all the eigenvalues of this matrix M are different.] [Remark: In this question, you must not state that it converges by using the power method.] e (d) Use the power method to solve Mw=wwith ɛ=0.1. e
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