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Possible Duplicate:
“Eigenrotations” of a matrix

have a question:

If a matrix $M$ acts by stretching a vector $x$ not changing its direction, then $x$ is an eigenvector of $M$.

Is there a complementary definition if $M$ acts on a vector $X$ not changing its magnitude (but changing its direction)?

Or are they considered equivalent definitions? It seems that we should be able to consider $x$ as an eigenvector in both cases.

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    Sure: they're called rotations.2012-05-06
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    There's a whole theory of antieigenvalue analysis (and a new book by Gustafson, its key developer) that considers the degree of rotation, but don't know if the magnitude is constrained to be invariant.2012-05-06
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    Well complex eigenvectors may help, they are really rotations. But it is another history!2012-05-06

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