Find the eigenvalues $\lambda_1 < \lambda_2 < \lambda_3$ and associated unit eigenvectors $\vec{u}_1, \vec{u}_2, \vec{u}_3$ of the symmetric matrix $$A = \left[\begin{array}{ccc} 3 &3 &3\cr 3 &-1 &7\cr 3 &7 &-1 \end{array}\right].$$
The eigenvalue $\lambda_1 =$ has associated unit eigenvector $\vec{u}_1 =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$.

The eigenvalue $\lambda_2 =$ has associated unit eigenvector $\vec{u}_2 =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$.

The eigenvalue $\lambda_3 =$ has associated unit eigenvector $\vec{u}_3 =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$.
Note: The eigenvectors above form an orthonormal eigenbasis for $A$.