Please read the italic part at the top of https://www.physicsforums.com/showthread.php?t=243852 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?