Let us take the following rotation matrix: (cos e - sin R = (1) sin 0 cos e a) Find the characteristic equation and show that the eigenvalues are e® and e-i. b) Find the normalized eigenvectors. For a two dimensional complex vector i = (a b)", where a, b are complex numbers, square of the norm of the vector i is || i ||²= |a|² + [b|?.

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,...
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Let us take the following rotation matrix:
cos 0 - sin
R =
(1)
sin 0 cos e
a) Find the characteristic equation and show that the eigenvalues are e and e-i.
b) Find the normalized eigenvectors. For a two dimensional complex vector ū = (a b)", where
a, b are complex umbers, square of the norm of the vector i is || i ||2= |a|² + [b|?.
c) Show that
R? – 2 cos 0R + I2 = 0.
(2)
This must be as dictated by Cayley-Hamilton theorem which we will prove in question 2.
d) Find the matrix P that diagonalizes R. That means find P such that
P
p-'RP = (
(3)
0 e-ie
e) Show that the determinant and the trace are similarity transformation invariant. That means
show that:
tr (P'RP) = tr R, and det (P-'RP) = det R
(4)
Transcribed Image Text:Let us take the following rotation matrix: cos 0 - sin R = (1) sin 0 cos e a) Find the characteristic equation and show that the eigenvalues are e and e-i. b) Find the normalized eigenvectors. For a two dimensional complex vector ū = (a b)", where a, b are complex umbers, square of the norm of the vector i is || i ||2= |a|² + [b|?. c) Show that R? – 2 cos 0R + I2 = 0. (2) This must be as dictated by Cayley-Hamilton theorem which we will prove in question 2. d) Find the matrix P that diagonalizes R. That means find P such that P p-'RP = ( (3) 0 e-ie e) Show that the determinant and the trace are similarity transformation invariant. That means show that: tr (P'RP) = tr R, and det (P-'RP) = det R (4)
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