= Symmetric part of an orthogonal matrix
{title2=$\lambda(\operatorname{sym}A)=\cos\theta\ \text{when}\ \lambda(A)=e^{i\theta}$}
For a real <orthogonal matrix> $A$, $A^{\mathsf T}=A^{-1}$. If $Av=\lambda v$ for a nonzero complex <eigenvector>, then $A^{\mathsf T}v=\lambda^{-1}v$. Therefore the same <eigenvector> has <eigenvalue> $(\lambda+\lambda^{-1})/2$ for the <symmetric part of a matrix> $(A+A^{\mathsf T})/2$. Orthogonality implies $|\lambda|=1$, so writing $\lambda=e^{i\theta}$ gives $\cos\theta$. The complete complex <orthonormal eigenbasis> of an <orthogonal matrix> shows that all the <eigenvalues> of its symmetric part lie in $[-1,1]$.
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