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Let us take the following rotation matrix: Cos e sin 0 R = sin cose). a) Find the characteristic equation and show that the eigenvalues are eto and e-i". b) Find the normalized eigenvectors. For a two dimensional complex vector = (a b)', where a, b are complex numbers, square of the norm of the vector is || i ||²= |a|2 + |6|2. c) Show that R? 2 cos 0R + I2 = 0 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-'RP -i0 e 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

Let us take the following rotation matrix: Cos e sin 0 R = sin cose). a) Find the characteristic equation and show that the eigenvalues are eto and e-i". b) Find the normalized eigenvectors. For a two dimensional complex vector = (a b)', where a, b are complex numbers, square of the norm of the vector is || i ||²= |a|2 + |6|2. c) Show that R? 2 cos 0R + I2 = 0 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-'RP -i0 e 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

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section: Chapter Questions
Problem 22RE
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Please answer the question (b) (c) (d) & (e)

Let us take the following rotation matrix: ( ) cos O – sin 0 R = sin 0 Cos O a) Find the characteristic equation and show that the eigenvalues are ei and e-i0. 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 || ||²= |a|2 + |b|². c) Show that R? - 2 cos 0R + I = 0. 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 eio p-'RP = ( e 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
Transcribed Image Text:Let us take the following rotation matrix: ( ) cos O – sin 0 R = sin 0 Cos O a) Find the characteristic equation and show that the eigenvalues are ei and e-i0. 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 || ||²= |a|2 + |b|². c) Show that R? - 2 cos 0R + I = 0. 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 eio p-'RP = ( e 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
l Airtel-Stay Home 9:41 PM @ 19% O a bartleby.com = bartleby E Q&A Math / Calculus / Q&A Library / cos 0 -sin 0 cos 0 R = sin 0 a) F... Cos 0 -sin 0 cos O R= sin 0 a) Find the chara... Characteristic equation = det A – 1I cos O – 1 – sin 0 sin 0 cos 0 – 1 = (cos 0 – 2)2 + sin? 0 = 2² – (2 cos 0)2 + cos² 0 + sin? 0 = 22 – (2 cos 0)2 + 1 For eigen values, we use quadratic formula 2 cos 0±/(2 cos 0)² –4 2 = cos 0 ± vcos? 0 – 1 2 = cos 0 ± V- sin? 0 1 = cos 0 + i sin 0 2 = e±i0 Was this solution helpful? 48
Transcribed Image Text:l Airtel-Stay Home 9:41 PM @ 19% O a bartleby.com = bartleby E Q&A Math / Calculus / Q&A Library / cos 0 -sin 0 cos 0 R = sin 0 a) F... Cos 0 -sin 0 cos O R= sin 0 a) Find the chara... Characteristic equation = det A – 1I cos O – 1 – sin 0 sin 0 cos 0 – 1 = (cos 0 – 2)2 + sin? 0 = 2² – (2 cos 0)2 + cos² 0 + sin? 0 = 22 – (2 cos 0)2 + 1 For eigen values, we use quadratic formula 2 cos 0±/(2 cos 0)² –4 2 = cos 0 ± vcos? 0 – 1 2 = cos 0 ± V- sin? 0 1 = cos 0 + i sin 0 2 = e±i0 Was this solution helpful? 48
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