Show that if \(U\) is an orthogonal matrix, then any real eigenvalue of \(U\) must be \( \pm 1\).

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 \( \pm 1\).