= Solution
For the $C$ obtained above, the <characteristic polynomial> is
$$
\det(C-\mu I)=(-4-\mu)(4-\mu)-9=\mu^2-25.
$$
Thus the <eigenvalues> are $5$ and $-5$. For $\mu=5$, $(C-5I)\mathbf v=0$ gives $v_2=-3v_1$; for $\mu=-5$, it gives $v_1=3v_2$. Normalizing the <eigenvectors> yields
$$
\boxed{\mu=5:\ \mathbf v_+=\frac1{\sqrt{10}}\begin{pmatrix}1\\-3\end{pmatrix};\qquad
\mu=-5:\ \mathbf v_-=\frac1{\sqrt{10}}\begin{pmatrix}3\\1\end{pmatrix}.}
$$
Both are <unit vectors>, their <dot product> is zero, and either overall sign is allowed for each <eigenvector>.
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