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

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

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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H4 part B
8:55 2.1. Eigenvalues and eigenvectors mtaylor.web.unc.edu Proof. 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) gives C₁(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 transformation A:R² R², A=(13¹) A₁2, A₂= 4, y = We can write 60 А We also display the circle x² + y² = 1, and its image under A. Compa Figure 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 are not real: (2.1.11) Ati√3, ¹₂: - 2/1/27 (1+i√³). 2√2 2 (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 m 59 2. Eigenvalues, eigenvectors, and generalized eigenvectors (5 17 Figure 2.1.1. Behavior of the linear transformation A in (2.1.8), with two distinct real cinemalnes
Transcribed Image Text:8:55 2.1. Eigenvalues and eigenvectors mtaylor.web.unc.edu Proof. 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) gives C₁(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 transformation A:R² R², A=(13¹) A₁2, A₂= 4, y = We can write 60 А We also display the circle x² + y² = 1, and its image under A. Compa Figure 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 are not real: (2.1.11) Ati√3, ¹₂: - 2/1/27 (1+i√³). 2√2 2 (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 m 59 2. Eigenvalues, eigenvectors, and generalized eigenvectors (5 17 Figure 2.1.1. Behavior of the linear transformation A in (2.1.8), with two distinct real cinemalnes
Log In Sep 12 Read your lecture notes and do Sec 2.3: 1 and watch the 61-minute video on companion matrices and rational canonical form (use the link provided or access it in the Modules section of our Canvas portal) and then do H4. (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 in Question 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 0 0 02 [2 1 01
Transcribed Image Text:Log In Sep 12 Read your lecture notes and do Sec 2.3: 1 and watch the 61-minute video on companion matrices and rational canonical form (use the link provided or access it in the Modules section of our Canvas portal) and then do H4. (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 in Question 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 0 0 02 [2 1 01
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