H4 part B 8:552.1. Eigenvalues and eigenvectorsmtaylor.web.unc.eduProof. We argue by contradiction. If {₁,...,} is linearly dependent,take a minimal subset that is linearly dependent and (reordering if necessary)say this set is (....,m), with Tej = Ajuy, and(2.1.6)C++-0,with c; 0 for each je {1,...,m). Applying T- AI to (6.6) givesC₁(A₁A)++¤m-1 (Am-1 - Am)¤m-1 = 0,(2.1.7)a linear dependence relation on the smaller set (1,..., -1}. This contra-diction proves the proposition.(2.1.8)with two distinct eigenvalues, and associated eigenvectors(2.1.9)-(1) - (-¹).See Figure 2.1.1 for an illustration of the action of the transformationA:R² R², A=(13¹)A₁2, A₂= 4, y =We can write60АWe also display the circle x² + y² = 1, and its image under A. CompaFigure 1.2.1.For contrast, we consider the linear transformation(2.1.10)A: R² R², A=(¹-1).whose eigenvalues A are purely imaginary and whose eigenvectors arenot real:(2.1.11)Ati√3, ¹₂:- 2/1/27 (1+i√³).2√22(2.1.12)and capture the behavior of A as(2.1.13)Aug =√√31, A = √30-See Figure 2.1.2 for an illustration. This figure also displays the ellipse(2.1.14)(t) = (cost)uo + (sin t)u₂, 0≤t≤ 2*,and its image under A.For another contrast, we look at the transformation(2.1.15)A:R² R², A=(1¹).to+im. 140=- 27/12 (2) 2/2 (3³).My m592. Eigenvalues, eigenvectors, and generalized eigenvectors(517Figure 2.1.1. Behavior of the linear transformation A in (2.1.8), withtwo distinct real cinemalnes Log InSep12Read your lecture notes and doSec 2.3: 1 andwatch the 61-minute video on companion matrices and rational canonical form (use the linkprovided or access it in the Modules section of our Canvas portal) and then doH4. (a) Compute the companion matrix for the polynomial found in Question 2.3.1.(b) Compute the companion matrix of the characteristic polynomial of the matrix given inQuestion 2.1.11.H5. (a) Suppose A and B are similar matrices. Show that AK = 0 iff Bk = 0, where k € N.(b) Find the characteristic polynomial of the matrix A4 given in H1.(c) Compute the companion matrix C of the polynomial found in (b).(d) Compute A4² and C².(e) Use (a) and (d) to show that A4 and C are NOT similar matrices.(H5(e) motivates using rational canonical form instead of C.)H6. Find the rational canonical form of each of the following matrices:(a) the matrix A4 given in H1[200](b) 0 2 00 02[2 1 01

Given: H4: b) as per the question 2.1.11 The given eigenvalues are λ=±i3 To find the companion…

Answered: -Fa CAFF-TY ZAZ (3^²+1) ²4² - ² We car…

-Fa CAFF-TY ZAZ (3^²+1) ²4² - ² We car (2.1.11) not real:

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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