= Solution
The <characteristic polynomial> is
$$
\det(A-\lambda I)=(5-\lambda)^2-9=(\lambda-2)(\lambda-8).
$$
For $\lambda=2$, the <eigenvector> equation reduces to $x=y$; for $\lambda=8$, it reduces to $x=-y$. Choose the following <unit vectors> consistently for the remaining parts:
$$
\boxed{\lambda_1=2,\quad x_1=\frac1{\sqrt2}\begin{pmatrix}1\\1\end{pmatrix};\qquad
\lambda_2=8,\quad x_2=\frac1{\sqrt2}\begin{pmatrix}1\\-1\end{pmatrix}.}
$$
Each <eigenspace> consists of all <scalar> multiples of its displayed <vector>, including zero; an <eigenvector> itself must be nonzero.
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