Eigenvalues of a complex orthogonal matrix

Question

• 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?

Answer 1

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 $-1$.