= Solution
The <characteristic polynomial> is $\det(\lambda I-M)=\lambda(\lambda^2-2)$. Solving the three <eigenvector> equations gives
$$
\boxed{\lambda=\sqrt2,\ 0,\ -\sqrt2;\qquad v_+=(1,\sqrt2,1)^T,\quad v_0=(1,0,-1)^T,\quad v_-=(1,-\sqrt2,1)^T.}
$$
Each <eigenspace> consists of the <scalar multiples> of its displayed nonzero <vector>. Their pairwise <dot products> vanish: $v_+\cdot v_-=1-2+1=0$ and $v_\pm\cdot v_0=1-1=0$. This also agrees with the <orthogonal diagonalization of a real symmetric matrix>.
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