= Solution
For real $\lambda_1,\lambda_2$, compute
$$
LL^T-L^TL=\begin{pmatrix}-1&\lambda_1-\lambda_2\\\lambda_1-\lambda_2&1\end{pmatrix}\ne0.
$$
Thus $L$ is a <non-normal matrix>. For distinct <eigenvalues>, take
$$
v_1=\begin{pmatrix}\lambda_1-\lambda_2\\1\end{pmatrix},\qquad v_2=\begin{pmatrix}0\\1\end{pmatrix}.
$$
These satisfy $Lv_j=\lambda_jv_j$ but have <inner product> $v_1^Tv_2=1$. The <eigenvectors> are therefore <nonorthogonal>. When the eigenvalues coincide, only the direction $v_2$ remains as an <eigenvector>; $L$ is a defective <Jordan block>.
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