Homework Help is Here – Start Your Trial Now!
Math
Algebra
2. In this course you mainly work with matrices that have real eigenvalues. But, what about the matrices that do not have only real eigenvalues? Is benefiting from eigenvalues and eigenvectors limited only to the matrices with real eigenvalues? Of course not. One may think working with complex numbers is difficult, however, utilizing complex numbers in many engineering problems simplifies the problem-solving process and gives us better insight into the problem itself. One simple example here cab be a rotation matrix. a. You know a 2D rotation matrix is cos e - sin 0 R that rotates the input vector x to sin 0 Cos O the vector Rx that is rotated version of vector x in 0 degrees counter-clockwise. We know that there is not a general real vector (real numbers), that maps Rx in the same direction of x. Because the characteristic polynomial of this matrix is cos 8 – A - sin 0 (cos 0 – 1)? + sin? 0 = X² – (2 cos 0)A +1= 0 %3D sin 0 cos 0 – A

2. In this course you mainly work with matrices that have real eigenvalues. But, what about the matrices that do not have only real eigenvalues? Is benefiting from eigenvalues and eigenvectors limited only to the matrices with real eigenvalues? Of course not. One may think working with complex numbers is difficult, however, utilizing complex numbers in many engineering problems simplifies the problem-solving process and gives us better insight into the problem itself. One simple example here cab be a rotation matrix. a. You know a 2D rotation matrix is cos e - sin 0 R that rotates the input vector x to sin 0 Cos O the vector Rx that is rotated version of vector x in 0 degrees counter-clockwise. We know that there is not a general real vector (real numbers), that maps Rx in the same direction of x. Because the characteristic polynomial of this matrix is cos 8 – A - sin 0 (cos 0 – 1)? + sin? 0 = X² – (2 cos 0)A +1= 0 %3D sin 0 cos 0 – A

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
See similar textbooks
icon
Related questions
Question
2. In this course you mainly work with matrices that have real eigenvalues. But, what about the matrices that do not have only real eigenvalues? Is benefiting from eigenvalues and eigenvectors limited only to the matrices with real eigenvalues? Of course not. One may think working with complex numbers is difficult, however, utilizing complex numbers in many engineering problems simplifies the problem-solving process and gives us better insight into the problem itself. One simple example here cab be a rotation matrix. a. You know a 2D rotation matrix is Cos O – sin 0 R sin 0 that rotates the input vector x to Cos O the vector Rx that is rotated version of vector x in 0 degrees counter-clockwise. We know that there is not a general real vector (real numbers), that maps Rx in the same direction of x. Because the characteristic polynomial of this matrix is - sin 0 cos 8 – A cos 8 – A (cos 0 – 1)2 + sin? 0 = x² – (2 cos 0)A+1=0 sin 0 Using algebra find the complex roots of the above second-degree characteristic polynomial. To do so you only need one trigonometric identity that is sin? 0+ cos? 0 = 1. b. Draw the locus of the eigenvalues you found in par' 1 in a complex plane. Indicate the location of roots for 0 = 30°, 45°, 90°. For which angles of 0 the eigenvalues of the rotation matrix are real (the imaginary part is zero)? Imaginarr
Transcribed Image Text:2. In this course you mainly work with matrices that have real eigenvalues. But, what about the matrices that do not have only real eigenvalues? Is benefiting from eigenvalues and eigenvectors limited only to the matrices with real eigenvalues? Of course not. One may think working with complex numbers is difficult, however, utilizing complex numbers in many engineering problems simplifies the problem-solving process and gives us better insight into the problem itself. One simple example here cab be a rotation matrix. a. You know a 2D rotation matrix is Cos O – sin 0 R sin 0 that rotates the input vector x to Cos O the vector Rx that is rotated version of vector x in 0 degrees counter-clockwise. We know that there is not a general real vector (real numbers), that maps Rx in the same direction of x. Because the characteristic polynomial of this matrix is - sin 0 cos 8 – A cos 8 – A (cos 0 – 1)2 + sin? 0 = x² – (2 cos 0)A+1=0 sin 0 Using algebra find the complex roots of the above second-degree characteristic polynomial. To do so you only need one trigonometric identity that is sin? 0+ cos? 0 = 1. b. Draw the locus of the eigenvalues you found in par' 1 in a complex plane. Indicate the location of roots for 0 = 30°, 45°, 90°. For which angles of 0 the eigenvalues of the rotation matrix are real (the imaginary part is zero)? Imaginarr
b. Draw the locus of the eigenvalues you found in part (a) in a complex plane. Indicate the location of roots for 0 = 30°, 45°, 90° . For which angles of 0 the eigenvalues of the rotation matrix are real (the imaginary part is zero)? Imaginary +i Real -1 C. Having the eigenvalues A1 and d2, find the complex eigenvectors of the rotation matrix. Note that the vector a where a and B are complex numbers and cos 0 – Ai - sin 0 +B Cos - is an sin 0 eigenvector for the eigenvalue of d;. Do eigenvectors of the rotation matrix depend on the angle of rotation 0?
Transcribed Image Text:b. Draw the locus of the eigenvalues you found in part (a) in a complex plane. Indicate the location of roots for 0 = 30°, 45°, 90° . For which angles of 0 the eigenvalues of the rotation matrix are real (the imaginary part is zero)? Imaginary +i Real -1 C. Having the eigenvalues A1 and d2, find the complex eigenvectors of the rotation matrix. Note that the vector a where a and B are complex numbers and cos 0 – Ai - sin 0 +B Cos - is an sin 0 eigenvector for the eigenvalue of d;. Do eigenvectors of the rotation matrix depend on the angle of rotation 0?
Expert Solution
steps

Step by step

Solved in 5 steps with 5 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Matrix Eigenvalues and Eigenvectors
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education
SEE MORE TEXTBOOKS