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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.

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    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.