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Math Help - Singular values

  1. #1
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    Singular values

    Let A be an orthogonal 2x2 matrix. Use the image of the unit circle to find the singular values of A.

    Seems like it should be an easy question but I don't understand how to do it...help please!
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  2. #2
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    Quote Originally Posted by Susaluda View Post
    Let A be an orthogonal 2x2 matrix. Use the image of the unit circle to find the singular values of A.

    Seems like it should be an easy question but I don't understand how to do it...help please!

    What do you mean by "singular values of a matrix", anyway?

    Tonio
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  3. #3
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    Singular values are what the chapter is on. It's a topic. I don't really know much about them though...

    Edit: The singular values are the square roots of the eigenvalues of the transpose of matrix A times A.
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    Quote Originally Posted by Susaluda View Post
    Singular values are what the chapter is on. It's a topic. I don't really know much about them though...

    Edit: The singular values are the square roots of the eigenvalues of the transpose of matrix A times A.

    Ok, so now we know what they are. Now, an orthogonal (real, of course...otherwise you'd call it unitary, right?) matrix is

    of the form A=\begin{pmatrix}\cos x&-\sin x\\\sin x&\cos x\end{pmatrix} , or A=\begin{pmatrix}\cos x&\sin x\\\sin x&-\cos x\end{pmatrix} ,where we can (not must) take 0\leq x\leq 2\pi

    Take for example the first kind, then A^tA=I=\begin{pmatrix}1&0\\0&1\end{pmatrix} , so the singular values are ... \pm 1 .

    Of course, this follows at once from the fact that A^t=A^{-1} for orthogonal (real) matrices....

    Tonio
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