<s>[INST] How do we find eigenvalues from given eigenvectors of a given matrix? For instance let 
$$A=\begin{pmatrix}
        3 & -1 & -1 \\
        2 & 1 &-2 \\
        0 & -1 & 2 \\
        \end{pmatrix}$$
be a matrix  and 
$$u_1=\begin{pmatrix}
        1 \\
        1  \\
        1 \\
        \end{pmatrix},$$
$$u_2=\begin{pmatrix}
        1 \\
        0 \\
        1\\
        \end{pmatrix},$$
$$u_3=\begin{pmatrix}
        0 \\
        -1 \\
        1 \\
        \end{pmatrix}.$$
its eigenvectors. What are its eigenvalues?
Is there anything more simple than doing $A-λI$?
 [/INST] If $u$ is an eigenvector of $A$ and $\lambda$ is the corresponding eigenvalue, you know the following:
$$Au = \lambda u$$
So in your example, you can do the following (I’ll take the second one):
$$Au_2 = \begin{pmatrix}3 & -1 & -1 \\ 2 & 1 & -2 \\ 0 & -1 & 2\end{pmatrix}\begin{pmatrix}1 \\ 0 \\ 1\end{pmatrix} = \begin{pmatrix}2 \\ 0 \\ 2\end{pmatrix} = 2 u_2$$
So $\lambda_2 = 2$. Do the same for $u_1$ and $u_3$.
</s>