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Is it true if we generalize to the complex case that an angle preserving transformation's eigenvalues are all the same magnitude? Is there a simple proof for the real case? Obviously its not a sufficient condition, but it is a necessary condition (see the link). But what's the proof.

This is how far I can get:

If its angle preserving, then for x,y eigenvectors (λ_x, λ_y eigenvalues) we get

(λ_x)(λ_y)°/|λ_x||λ_y| = (λ_x)°(λ_y)/|λ_x||λ_y| = 1 where ° denotes the complex conjugate.

But I'm not sure where to go from here..... is there a simple contradiction that can show itself?

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# Spivak correction?

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