Showing that the limit of non-eigenvector goes to infinity Let $A$ be a $3$ by $3$ real matrix with the triple eigenvalue $1$. Also, further suppose its eigenspace corresponding to $1$ is only of dimension $1$. Thus, we can find a basis of $\mathbb{R}^3$, denoted by $v$. $w_1$. $w_2$ where $v$ is an eigenvector of $A$.
Then I have to show that $\lim_{n \to \infty} \|A^n w_1\|=\lim_{n \to \infty} \|A^n w_2\|=\infty$. 
How is this possible? I do not have any idea how the norm goes to infinity...Could anyone please help me?

The theorem is correct. Without loss of generality and by using Schur decomposition, we can write the most general form of matrix $A$ as$$A=\begin{bmatrix}1&a&b\\0&1&c\\0&0&1\end{bmatrix}$$Since the eigenspace of $A$ has dimension $1$, and the eigenvector is $v=\begin{bmatrix}v_1\\v_2\\v_3\end{bmatrix}$ therefore the following set of equations$$\begin{bmatrix}1&a&b\\0&1&c\\0&0&1\end{bmatrix}\begin{bmatrix}v_1\\v_2\\v_3\end{bmatrix}=\begin{bmatrix}v_1\\v_2\\v_3\end{bmatrix}$$must set two of $\{v_1,v_2,v_3\}$ equal to zero. Since the set can be reduced to$$av_2+bv_3=0\\cv_3=0$$therefore we can have $v_1\ne0$ and $v_2=v_3=0$. The only case this happens is when$$a,c\ne 0$$. Therefore the only eigenvector of $A$ becomes $$\begin{bmatrix}1\\0\\0\end{bmatrix}$$Now take $w_1=\begin{bmatrix}0\\w_{12}\\w_{13}\end{bmatrix}$ and $w_2=\begin{bmatrix}0\\w_{22}\\w_{23}\end{bmatrix}$ where $w_2\ne kw_1$. One can prove using induction that$$A^nw_1=\begin{bmatrix}naw_{12}+nbw_{13}+{n(n-1)\over 2}acw_{13}\\w_{12}+ncw_{13}\\w_{13}\end{bmatrix}\\A^nw_2=\begin{bmatrix}naw_{22}+nbw_{23}+{n(n-1)\over 2}acw_{23}\\w_{22}+ncw_{23}\\w_{23}\end{bmatrix}$$Since $a,c\ne 0$, it is easy to check that$$||A^nw_1||\to \infty\\||A^nw_2||\to \infty$$$\blacksquare$
P.S.
The general form of $A$ has been chosen upper-triangular since from the Schur decomposition and for any arbitrary vector $v$ we have $$||A^nv||{=||QU^nQ^Hv||\\=||QU^nw||\\=||U^nw||}$$where $w\triangleq Q^Hv$ and $U$ and $Q$ are upper-triangular and unitary respectively.
