Context : Let's suppose $L$ is a linear map from $\mathbb{R^k}\rightarrow \mathbb{R}^k$ , $k$ strictly positive integer. Let's suppose $\epsilon$ is a strictly positive real.
In an exercice , i have to show that there exists a basis $\mathcal{B}$ of $\mathbb{C}^k $ such that , $L$ has a upper triangular representation and that in this basis , every coefficient of the matrix representation of L is strictly inferior to $\epsilon$.
I want to say : In $\mathbb{C}^k$ , caracteristic polynomial of L is splitable , a basis in $\mathbb{C}^k$ in which L is upper triangular exists. In particular , there exists a basis $(a_i)_i$ in which L can be written in jordan form. (see https://en.wikipedia.org/wiki/Jordan_normal_form for a definition i guess)
But i can't figure out how to find a particular one respecting "very coefficient of the matrix representation of L is strictly inferior to $\epsilon$.". I found a clue that says that the basis $b_i=\epsilon^{i-1}a_i$ $i \in [0,k]$) works ,but i don't really figure out why,can someone explain ? or correct,
edit : thanks Spencer Leslie for the comment , i think i've been neglecting the hypothesis : the correction i make on the coefficient is i want $\vert m_{i,j}\vert <\epsilon \ \forall i,j,\ 1 \leq i<j\leq k$ (every non diagonal coefficient)
Edit 2 : correcting some terminology mistakes.