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Question 02 Consider the following matrix: cos 0 – sin 0 R = - sin 0 - cos 0 a) Find the eigenvalues A1, A2 of R. Hint: Eigenvalues are the solution to det (R – AI) = = 0. b) Show that cos (0/2) sin (0/2) sin (0 U2 = cos (0/2) are the unit eigenvectors of R with eigenvalues –1 and 1 respectively. That means prove that (sin (0/2) ) cos (8/2) ´sin (0/2) cos (0/2), - sin 0 cos e sin 0 and - cos e - sin 0 cos e - sin 0 - cos e cos (0/: sin (0/2) Cos sin (0/2) Hint: You may find these formulae useful: sin (a + B) = sin a cos 3± cos a sin 3, cos (a ± B) = cos a cos BF sin a sin 3. c) Find the matrix P that diagonalizes R. That means find P such that P-'RP = D. Find D. Hint: P (vi 02) or P = (02 v1), and D is the diagonal matrix with the correspondin eigenvalues along the diagonal.

Question 02 Consider the following matrix: cos 0 – sin 0 R = - sin 0 - cos 0 a) Find the eigenvalues A1, A2 of R. Hint: Eigenvalues are the solution to det (R – AI) = = 0. b) Show that cos (0/2) sin (0/2) sin (0 U2 = cos (0/2) are the unit eigenvectors of R with eigenvalues –1 and 1 respectively. That means prove that (sin (0/2) ) cos (8/2) ´sin (0/2) cos (0/2), - sin 0 cos e sin 0 and - cos e - sin 0 cos e - sin 0 - cos e cos (0/: sin (0/2) Cos sin (0/2) Hint: You may find these formulae useful: sin (a + B) = sin a cos 3± cos a sin 3, cos (a ± B) = cos a cos BF sin a sin 3. c) Find the matrix P that diagonalizes R. That means find P such that P-'RP = D. Find D. Hint: P (vi 02) or P = (02 v1), and D is the diagonal matrix with the correspondin eigenvalues along the diagonal.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question 02 Consider the following matrix: cos 0 – sin 0 R= - sin 0 - cos 0 a) Find the eigenvalues A1, X2 of R. Hint: Eigenvalues are the solution to det (R – AI) = = 0. b) Show that (). - cos (0/2) sin (0/2) sin (0 T2 = cos (0/2) are the unit eigenvectors of R with eigenvalues –1 and 1 respectively. That means prove that: ´sin (0/2) ) cos (8/2) cos (0/: sin (0/2) cos e sin 0 - sin 0 - cos e ´sin (0/2) cos (0/2), and cos e - sin 0 cos (0/2) - sin 0 - cos 0 sin Hint: You may find these formulae useful: sin (a + B) = sin a cos 3± cos a sin 3, cos (a ± B) = cos a cos B7 sin a sin 3. c) Find the matrix P that diagonalizes R. That means find P such that P-'RP = D. Find D. Hint: P (vi 02) or P = (02 v1), and D is the diagonal matrix with the corresponding eigenvalues along the diagonal.
Transcribed Image Text:Question 02 Consider the following matrix: cos 0 – sin 0 R= - sin 0 - cos 0 a) Find the eigenvalues A1, X2 of R. Hint: Eigenvalues are the solution to det (R – AI) = = 0. b) Show that (). - cos (0/2) sin (0/2) sin (0 T2 = cos (0/2) are the unit eigenvectors of R with eigenvalues –1 and 1 respectively. That means prove that: ´sin (0/2) ) cos (8/2) cos (0/: sin (0/2) cos e sin 0 - sin 0 - cos e ´sin (0/2) cos (0/2), and cos e - sin 0 cos (0/2) - sin 0 - cos 0 sin Hint: You may find these formulae useful: sin (a + B) = sin a cos 3± cos a sin 3, cos (a ± B) = cos a cos B7 sin a sin 3. c) Find the matrix P that diagonalizes R. That means find P such that P-'RP = D. Find D. Hint: P (vi 02) or P = (02 v1), and D is the diagonal matrix with the corresponding eigenvalues along the diagonal.
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