Recall that a rotation matrix Re (sind) induces a linear trans- sin 0 formation which rotates...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Recall that a rotation matrix Re (sind) induces a linear trans- sin 0 formation which rotates a vector in R2 by an angle 0 ≤ 0 < 2 in the counter-clockwise. = (1) Show that det Re 1 for any 0. = cos - sin Let SO(2, R) {A € M₂(R) | A-¹ = At and det(A) = 1} be the set of special orthogonal matrices. Show that if A E SO(2, R), then A = Ro equals to a rotation matrix for some 0 ≤ 0 < 2. = (3) Show that the rotation matrix R/2 has no real eigenvalue. (4) Find all the possible values for 0 ≤ 0 < 2π such that the rotation matrix Re has a real eigenvalue. If it is the case, find all the real eigenvalues and corresponding eigenvectors (in R²). Recall that a rotation matrix Re (sind) induces a linear trans- sin 0 formation which rotates a vector in R2 by an angle 0 ≤ 0 < 2 in the counter-clockwise. = (1) Show that det Re 1 for any 0. = cos - sin Let SO(2, R) {A € M₂(R) | A-¹ = At and det(A) = 1} be the set of special orthogonal matrices. Show that if A E SO(2, R), then A = Ro equals to a rotation matrix for some 0 ≤ 0 < 2. = (3) Show that the rotation matrix R/2 has no real eigenvalue. (4) Find all the possible values for 0 ≤ 0 < 2π such that the rotation matrix Re has a real eigenvalue. If it is the case, find all the real eigenvalues and corresponding eigenvectors (in R²).
Expert Answer:
Related Book For
Cambridge International AS And A Level Mathematics Pure Mathematics 1 Coursebook
ISBN: 9781108407144
1st Edition
Authors: Sue Pemberton, Julian Gilbey
Posted Date:
Students also viewed these mathematics questions
-
The 2 à 2 matrix is called a rotation matrix because y = QX is the rotation of x by the angle θ. Suppose X = [X1 X2] is a Gaussian (0, CX) vector where CX = diag[Ï21,...
-
In Exercise, show that the given transformation from R2 to R2 is linear by showing that it is a matrix transformation a. F reflects a vector in the y-axis. b. R rotates a vector 45 counterclockwise...
-
Show that f in example 3.1 is linear. The function f : 2 2 defined by F( x1, x2) = (x1 cos - x2 sin ,x1 sin + x2 cos ), 0 0 2 rotates any vector in the plane counter clock wise through the angle...
-
How do recruitment and selection practices contribute to high performance in an organization?
-
What is the conversion factor between? (a) ft2 and yd2 (a) m2 and ft2?
-
Derive the equation MC = IC for the rolling disk of Fig. 16.17, where MC represents the sum of the moments of the external forces about the instantaneous center C, and IC is the moment of inertia of...
-
Alpine Ski Shops Prepaid Rent balance is $4,500 on June 1. This prepaid rent rep resents six months rent. Journalize and post the adjusting entry on June 30 to record one months rent. Compute the...
-
Santana Rey, owner of Business Solutions, realizes that she needs to begin accounting for bad debts expense. Assume that Business Solutions has total revenues of $44,000 during the first three months...
-
Age: 25 Filing status: Single Education: Bachelor's degree obtained 3 years ago Job: Senior Analyst for New York Financial Corporation Salary: $75,000 per year (before taxes and deductions);...
-
Determine the output voltage for an integrator with a single-pulse input, as shown in Figure 1556. For repetitive pulses, how long will it take this circuit to reach steady state? 8 V 0 1 ms O...
-
1. Describe a situation that you encounter in your own life that can be repre- sented as a game of complete information 2. Describe a second, and different situation that you encounter in your own...
-
When a commercial property is sold and has a tenant in occupations of the property, what is the name of the document to be provided to the tenant to acknowledge the new owner of the premises? b) To...
-
How does transformational leadership influence organizational culture, and what mechanisms enable such leaders to effect deep-seated cultural change ?
-
what ways does social control theory offer insights into the mechanisms that maintain conformity and deter deviance within complex social structures ?
-
How do labeling theories explain the process by which individuals or groups come to be classified as deviant, and what are the long-term consequences of such labeling ?
-
January 5th Problem: Traded in all of used computer equipment with a fair value of $13,200,000., and received in exchange new computer equipment (useful life: 10 years, salvage value: 1,947,040) with...
-
Discuss the essential characteristics an entrepreneur must possess to be successful in global negotiations, in addition to technical ability. Explain the significance of each characteristic in the...
-
Research an article from an online source, such as The Economist, Wall Street Journal, Journal of Economic Perspectives, American Journal of Agricultural Economics, or another academic journal. The...
-
The line l1 has equation 3x 2y = 12 and the line l 2 has equation y = 15 2x. The lines l 1 and l 2 intersect at the point A. a. Find the coordinates of A. b. Find the equation of the line through A...
-
Find the value of x, correct to 3 significant figures. Rectangle A has sides of length x cm and (2x 4) cm. Rectangle B has sides of length (x + 1) cm and (5 x) cm. Rectangle A and rectangle B have...
-
Calculate the length of QR. P 1 rad 5 cm R
-
What is the IQR? What does the IQR represent? The population parameters below describe the full-time equivalent number of students (FTES) each year at Lake Tahoe Community College from 19761977...
-
How many standard deviations away from the mean is the median? The population FTES for 20052006 through 20102011 was given in an updated report. The data are reported here. The population parameters...
-
A class of 20 students has a mean grade of 80 on a test. Nineteen of the students has a mean grade between 79 and 82, inclusive. a. What is the lowest possible grade of the other student? b. What is...
Study smarter with the SolutionInn App