A unitary matrix in which all entries are real is the same thing as an orthogonal matrix. Just as an orthogonal matrix G preserves the (real) inner product of two real vectors, thus
A matrix is unitary if and only if its columns form an orthonormal basis of C^{n} with respect to this inner product.
All eigenvalues of a unitary matrix are complex numbers of absolute value 1, i.e. they lie on the unit circle centered at 0 in the complex plane. The same is true for its determinant.
All unitary matrices are normal, and the spectral theorem therefore applies to them.
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