# Orthogonal and unitary operators

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?

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