$x, Ax, \cdots, A^mx, \cdots$ is not dense in $\Bbb R^n$, where $A$ is a $n\times n$ real matrix, $x\in\Bbb R^n$.

Question

$x, Ax, \cdots, A^mx, \cdots$ is not dense in $\Bbb R^n$, where $A$ is a $n\times n$ real matrix, $x\in\Bbb R^n$ are fixed.

This is a problem involving mathematical analysis and higher algebra. I have no idea.

1. Would we use the Jordan carnonical form, but the imaginary part seems trouble.

2. What about some orthogonal matrix $Q$ exists, such that $Q^{-1}AQ$ is an upper block triangular matrix.