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Theorem 7states that consider that \(U\) as an \(m \times n\) matrix with orthonormal columns and assume that x and y are in \({\mathbb{R}^n}\). Then;

1. \(\left\| {U{\bf{x}}} \right\| = \left\| {\bf{x}} \right\|\)
2. \(\left( {U{\bf{x}}} \right) \cdot \left( {U{\bf{y}}} \right) = {\bf{x}} \cdot {\bf{y}}\)
3. \(\left( {U{\bf{x}}} \right) \cdot \left( {U{\bf{y}}} \right) = 0\)such that if \({\bf{x}} \cdot {\bf{y}} = 0\).

When \(U{\bf{x}} = \lambda {\bf{x}}\) for some \({\bf{x}} \ne 0\), then according to Theorem 7 and by a property of the norm:

\(\begin{array}{c}\left\| {\bf{x}} \right\| = \left\| {U{\bf{x}}} \right\|\\ = \left\| {\lambda {\bf{x}}} \right\|\\ = \left| \lambda \right|\left\| {\bf{x}} \right\|\end{array}\)

This demonstrates that \(\left| \lambda \right| = 1\), since \({\bf{x}} \ne 0\).

Thus, it is proved that any real eigenvalues of \(U\) is \( \pm 1\).