ch6 - CHAPTER 6 EIGENVALUES AND EIGENVECTORS SECTION 6.1...

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Section 6.1289CHAPTER 6EIGENVALUES AND EIGENVECTORSSECTION 6.1INTRODUCTION TO EIGENVALUESIn each of Problems 1–32 we first list the characteristic polynomial( )pλλ=AIof the givenmatrixA, and then the roots of( )pλ— which are the eigenvalues ofA.All of the eigenvaluesthat appear in Problems 1–26 are integers, so each characteristic polynomial factors readily.Foreach eigenvaluejλof the matrixA,we determine the associated eigenvector(s) by finding a basisfor the solution space of the linear system().jλ=AI v0We write this linear system in scalarform in terms of the components of[].Tab=vIn most cases an associated eigenvector isthen apparent.IfAis a22×matrix, for instance, then our two scalar equations will be multiplesone of the other, so we can substitute a convenient numerical value for the first componentaofvand then solve either equation for the second componentb(or vice versa).1.Characteristic polynomial:2( )56(2)(3)pλλλλλ=+=Eigenvalues:122,3λλ==With12:λ=2200abab==111 =  vWith23:λ=2020abab==221 =  v2.Characteristic polynomial:2( )2(1)(2)pλλλλλ==+Eigenvalues:121,2λλ= −=With11:λ= −660330abab==111 =  vWith22:λ=360360abab==221 =  v
290Chapter 63.Characteristic polynomial:2( )710(2)(5)pλλλλλ=+=Eigenvalues:122,5λλ==With12:λ=660330abab==111 =  vWith25:λ=360360abab==221 =  v4.Characteristic polynomial:2( )32(1)(2)pλλλλλ=+=Eigenvalues:122,5λλ==With11:λ=330220abab==111 =  vWith22:λ=230230abab==232 =  v5.Characteristic polynomial:2( )54(1)(4)pλλλλλ=+=Eigenvalues:121,4λλ==With11:λ=990660abab==111 =  vWith24:λ=690690abab==232 =  v6.Characteristic polynomial:2( )56(2)(3)pλλλλλ=+=Eigenvalues:121,4λλ==With12:λ=440330abab==111 =  vWith23:λ=340340abab==243 =  v7.Characteristic polynomial:2( )68(2)(4)pλλλλλ=+=Eigenvalues:122,4λλ==
Section 6.1291With12:λ=880660abab==111 =  vWith24:λ=680680abab==243 =  v8.Characteristic polynomial:2( )32(2)(1)pλλλλλ=++=++Eigenvalues:122,1λλ= −= −With12:λ= −9601280abab==123 =  vWith21:λ= −8601290abab==234 =  v9.

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Term
Spring
Professor
Zhu
Tags
Differential Equations, Linear Algebra, Eigenvectors, Equations, Vectors, Matrices, Eigenvalues, Characteristic polynomial, Eigenvalue eigenvector and eigenspace, Orthogonal matrix

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