Find the eigenvalues $\lambda_1 < \lambda_2$ and associated unit eigenvectors $\vec{u}_1, \vec{u}_2$ of the symmetric matrix $$A = \left[\begin{array}{cc} -1 &8\cr 8 &11 \end{array}\right].$$
The smaller eigenvalue $\lambda_1 =$ has associated unit eigenvector $\vec{u}_1 =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$.

The larger eigenvalue $\lambda_2 =$ has associated unit eigenvector $\vec{u}_2 =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$.
Note: The eigenvectors above form an orthonormal eigenbasis for $A$.