Eigenvalue of a rotation matrix

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Homework Help Overview

The discussion revolves around finding the eigenvalues of a 2D rotation matrix represented by the cosine and sine functions. Participants are exploring the characteristics of this matrix and the methods to derive its eigenvalues.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster seeks guidance on how to initiate the process of finding eigenvalues, expressing difficulty with standard methods. Other participants inquire about the progress made and suggest solving the characteristic polynomial as a potential approach.

Discussion Status

The discussion is active, with participants providing insights into the nature of the rotation matrix and its properties. There is mention of the relationship between unitary operators and eigenvalues, indicating a productive exploration of the topic.

Contextual Notes

Participants are considering the implications of the matrix being unitary and the properties of its eigenvalues, specifically their absolute values. There is an acknowledgment of the complexity involved in the problem, particularly regarding the use of the quadratic formula for complex eigenvalues.

supermesh
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cos a -sin a

sin a cos a

How do I find the eigenvalue of this rotation matrix? I did the usual way, but didn't work! Could someone tell me how to start this problem?
 
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How far did you get doing it the 'usual way'? It should work.
 
Did you try solving the characteristic polynomial?
 
A rotation matrix in C^n is unitary. Unitary linear operators have eigenvalues with absolute value 1 (because unitary transformations are also normal). You will get two complex eigenvalues, both with absolute value 1. Use the high school complex quadratic formula.
 
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