<s>[INST] Eigen value and eigen vectors I have given a matrix $$ \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & -3 & 3 \end{pmatrix}. $$ 
We have to find 3 linearly independent eigen vector. I have calculated the eigen values that is 1 with multiplicity 3 . I have found the eigen  vector $ \begin{pmatrix}1 & 1 & 1 \end{pmatrix}^T$. But I can't find the other two linearly independent eigen vectors. 
Any help is appreaciating.
 [/INST] The characteristic polynomial is
\begin{align}
\det\begin{pmatrix}
0-X & 1 & 0 \\
0 & 0-X & 1 \\
1 & -3 & 3-X
\end{pmatrix}
&=
-X\det\begin{pmatrix}-X & 1 \\ -3 & 3-X\end{pmatrix}
-\det\begin{pmatrix}0 & 1 \\ 1 & 3-X\end{pmatrix}
\\
&=-X(-3X+X^2+3)+1
\\[6px]
&=-X^3+3X^2-3X+1=(1-X)^3
\end{align}
There cannot exist three linearly independent vectors: the matrix would be diagonalizable and similar to
$$
\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}
$$
but the identity matrix is only similar to itself.
Actually, if we compute the rank of $A-I$ (where $A$ is the given matrix), we get, with Gaussian elimination,
$$
A-I=\begin{pmatrix}
-1 & 1 & 0 \\
0 & -1 & 1 \\
1 & -3 & 2
\end{pmatrix}
\to
\begin{pmatrix}
-1 & 1 & 0 \\
0 & -1 & 1 \\
0 & -2 & 2
\end{pmatrix}
\to
\begin{pmatrix}
-1 & 1 & 0 \\
0 & -1 & 1 \\
0 & 0 & 0
\end{pmatrix}
$$
so the rank is $2$ and the eigenspace relative to $1$ has dimension $3-2=1$. No set of two eigenvalues can be linearly independent.
</s>