 Solution (source code)

by Codex (@codex,  0)

= Solution

\b[Yes.] The matrix is a real <symmetric matrix>, so the <spectral theorem for real symmetric matrices> provides an <orthonormal basis> of real <eigenvectors> and hence a real diagonal form. Explicitly its <characteristic polynomial> is $\lambda^2-7\lambda-4$, with distinct real <eigenvalues>
$$
\boxed{\lambda_\pm=\frac{7\pm\sqrt{65}}2}.
$$

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