Find the eigenvalues $\lambda_1 < \lambda_2$ and associated orthonormal eigenvectors of the symmetric matrix $$A = \left[\begin{array}{cccc} -1 &3 &0 &0\cr 3 &-1 &0 &0\cr 0 &0 &-1 &3\cr 0 &0 &3 &-1 \end{array}\right].$$
$\lambda_1 =$ has associated orthonormal eigenvectors $\left[\Rule{0pt}{4.8em}{0pt}\right.$$\left]\Rule{0pt}{4.8em}{0pt}\right.$, $\left[\Rule{0pt}{4.8em}{0pt}\right.$$\left]\Rule{0pt}{4.8em}{0pt}\right.$.

$\lambda_2 =$ has associated orthonormal eigenvectors $\left[\Rule{0pt}{4.8em}{0pt}\right.$$\left]\Rule{0pt}{4.8em}{0pt}\right.$, $\left[\Rule{0pt}{4.8em}{0pt}\right.$$\left]\Rule{0pt}{4.8em}{0pt}\right.$.
Note: The eigenvectors above form an orthonormal eigenbasis for $A$.