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# Matrix Theory - Isometries

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Find the eigenvalues and eigenvectors of A.
See attached file for full problem description.

Consider the matrices in O(2) of the form:

;

these matrices correspond to the elements of O(2) with det(A)= -1. Find the eigenvalues and eigenvectors of A.

https://brainmass.com/math/linear-algebra/matrix-theory-isometries-2405

#### Solution Preview

I will use t instead of theta, it is easier to write; to find the eigenvalues we need to solve Det(A-xI)=0, which means:
(cost-x)(-cost-x)-sint*sint=0 so:
x^2=1 (using that the cos^2+sin^2=1).
So x1=1 and x2=-1 are the eigenvalues.
In what follows we will assume sint non-zero. If sint is zero, we discuss in the end of the proof what the answer is.
Eigenvectors for ...

#### Solution Summary

This shows how to find eigenvalues and eigenvectors.

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