Let $A=[a_{ij}]$ be a matrix of type $n\times n$ with coefficients with the field $K$, where $K=\mathbb{R}$ or $K=\mathbb{C}$.
In the case $K=\mathbb{C}$, for example by Jordan decomposition theorem, there exist an invertible matrice $B$ with coefficients from $K=\mathbb{C}$ such that $C:=B^{-1}AB$ is a upper triangular matrix, i.e $C=[c_{ij}]$ with $c_{ij}=0$ for $i>j$.
Is it true that for every matrice $A$ of type $n\times n$ with real coefficients, which characteristic polynomial has $n$ real roots not necessarily different, there exists an invertible matrix $B$ of the same type with real coefficients such that $B^{-1}AB$ is upper triangular?
Thanks.