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1) Theorem:

Let $T$ be a linear operator on a finite-dimensional real inner product space $V$. Then $V$ has an orthonormal basis of eigenvectors of $T$ with corresponding eigenvalues of absolute value 1 if and only if $T$ is both self adjoint and orthogonal.

2) Theorem:

Let $T$ be a linear operator on a finite-dimensional complex inner product space $V$. Then $V$ has an orthonormal basis of eigenvectors with absolute value 1 if and only if $T$ is unitary.

So my question is why doesnt a unitary operator have to be self-adjoint to have orthonormal basis of eigenvectors with corresponding eigenvalues of absolute value 1 but orthogonal operators have to?

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  • $\begingroup$ Orthogonal operators could still be missing eigenvectors since they may have complex eigenvalues, for example a 90 degree rotation of a plane. $\endgroup$ – Joppy Mar 14 at 21:13

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