= Solution
The third coordinate is decoupled, giving the unit <eigenvector> $(0,0,1)^{\mathsf T}$ with <eigenvalue> $1$. In the first two coordinates, the symmetric direction $(1,1)$ has <eigenvalue> $3-1=2$, and the antisymmetric direction $(1,-1)$ has <eigenvalue> $3+1=4$. Their normalized forms are
$$
\boxed{\begin{array}{c|c}
\lambda_1=2&\mathbf e_1=\dfrac1{\sqrt2}(1,1,0)^{\mathsf T}\\[3pt]
\lambda_2=4&\mathbf e_2=\dfrac1{\sqrt2}(1,-1,0)^{\mathsf T}\\[3pt]
\lambda_3=1&\mathbf e_3=(0,0,1)^{\mathsf T}.
\end{array}}
$$
They are an <orthonormal basis> of $\mathbb R^3$; independent sign reversals are equally valid choices.
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