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  • Property 1:A complex orthogonal matrix must have eigenvalues with modulus 1.
  • Property 2: If all entries in the matrix are real (real orthogonal matrix), then the eigenvalues must be $\pm 1$

Proof of Propery 2 Let $\lambda$ be an eigenvalue of $A$. Then, for some $\vec{x}\neq\vec{0}$, we must have: $$A\vec{x}=\lambda\vec{x}$$

Take the tranpose of both sides, and use the tranpose property to get: $$(\vec{x})^TA^T=\lambda(\vec{x})^T$$

Post-multiply by $A\vec{x}=\lambda\vec{x}$ on both sides: $$(\vec{x})^TA^TA\vec{x}=\lambda(\vec{x})^T\lambda\vec{x}$$

Since $A$ is orthogonal, substitute $A^TA=I$ on the LHS, note the $I$ "vanishes", and rearrange the scalars on the RHS: $$(\vec{x})^T\vec{x}=\lambda^2(\vec{x})^T\vec{x}$$

Compare both sides to get (for real or complex $\lambda$): $$\lambda^2=1 \implies \lambda=\pm 1$$

Question Which part of the proof above (correct as per my knowledge for real orthogonal matrices) fails for complex orthogonal matrices?

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  • $\begingroup$ Matrices must be orthonormal to have eigenvalues of modulus 1. $\endgroup$ Commented Mar 19 at 15:55
  • $\begingroup$ @quinzio Aren't orthogonal matrices also orthonormal. en.wikipedia.org/wiki/Orthogonal_matrix $\endgroup$ Commented Mar 19 at 17:26
  • $\begingroup$ It's a matter of definition. but it looks correct. $\endgroup$ Commented Mar 19 at 17:32

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Property 1 is false. See my here for instance.

Both your property 2 and your proof of it are also wrong. E.g. every 2x2 rotation matrix $A\ne\pm I$ has non-real eigenvalues. However, assuming that a real orthogonal matrix $A$ has a real eigenvalue $\lambda$, then your argument is correct and $\lambda$ is indeed equal to $1$ or $-1$.

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  • $\begingroup$ Can you tell where the argument fails for matrices with non real eigenvalues? $\endgroup$ Commented Mar 19 at 17:27
  • $\begingroup$ @Starlight When an eigenvalue is non-real, its associated eigenvector $\vec{x}$ must be non-real too. Therefore $(\vec{x})^T\vec{x}$ can be zero. $\endgroup$ Commented Mar 19 at 17:31
  • $\begingroup$ Why did you convert to a community wiki? $\endgroup$ Commented Mar 19 at 17:34

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