Understanding Eigenvalues in Rotational Transformations: A False Assertion

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SUMMARY

The assertion that a rotation Tθ in the Euclidean plane R² can be represented by an orthogonal matrix P with eigenvalues λ1 = 1 and λ2 = -1 is false. Rotations are indeed linear transformations and can be represented by matrices, but the eigenvalues of a rotation matrix are complex numbers, specifically e^(iθ) and e^(-iθ), not real values. The confusion arises from the misunderstanding of how rotations interact with eigenvectors and the properties of orthogonal matrices.

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Homework Statement



True/False

If Ttheta is a rotation of the Euclidean plane R2 counterclockwise through an angle theta, then T can be represented by an orthogonal matrix P whose eigenvalues are lambda1 = 1 and lambda2 = -1.

Homework Equations


The Attempt at a Solution



Just checking to see if my thinking is right. I say false because the representation of T in orthogonal coordinates would require a transformation requiring trigonometric functions. This wouldn't be a linear transformation and therefore cannot be treated as an eigensystem.

Yes? No?
 
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The answer is false, but I can't figure out what you're trying to say for your reasoning. Rotations are linear functions of the plane, and can be represented my matrices. You can either consider determinants, or try to demonstrate no rotation can do what is proposed to the two eigenvectors
 


Using sines and cosines is still a linear transformation.

Think of it this way, what is the determinant of a matrix such that it's eigenvalues are -1,1?
 

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