Proof for Angle Preserving Transformation: Eigenvalues Same Magnitude

[brydustin]

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?

 

[brydustin]

...okay, no need for an explanation. I found a proof.
http://christianmarks.wordpress.com/2009/07/06/spivaks-botched-problem/

 

[math771]

Hi there! Yes, it is true that for a complex angle-preserving transformation, the eigenvalues must have the same magnitude. This can be proven using the fact that an angle-preserving transformation is also orthogonally diagonalizable, meaning that it can be written as T = PDP^-1, where P is an orthogonal matrix and D is a diagonal matrix with the eigenvalues on the diagonal.

Since T is angle-preserving, we know that P must be a rotation matrix, which means that it preserves lengths and angles. This implies that P^-1 must also be a rotation matrix.

Now, let's consider the product P^-1DP. This is equivalent to multiplying a rotation matrix (P^-1) by a diagonal matrix (D), and then multiplying the result by another rotation matrix (P). Since rotations preserve lengths and angles, this product must also preserve lengths and angles.

But we know that the diagonal elements of D are the eigenvalues of T, and since the product P^-1DP preserves lengths and angles, this means that the eigenvalues must have the same magnitude.

I hope this helps! Let me know if you have any further questions.