Let $K$ be a field, $n\ge 1\in\mathbb{N}_{\ge 1}$ and let $A,B\in M_n(K)$ be two triagonalizable matrices, such that for every eigenvalue $\lambda$ of $A$ there holds that $\dim V(\lambda,A)^{\text{all}}\le 3$, where $V(\lambda,A)^{\text{all}}$ denotes the according generalized eigenspace.
Show that $A$ and $B$ are similar iff $\chi_A=\chi_B$ and $\mu_A=\mu_B$, where $\chi_{.}$ and $\mu_{.}$ denote the characteristic and minimal polynomial, respectively.
From the above assumptions I can see that $A,B$ have invariant flags and the dimension of the generalized eigenspaces gives us information of the Jordan normal form of $A$. How do I use this to prove the claim?
Fix a single eigenvalue $\lambda$ and inspect the possible Jordan blocks referred to $\lambda$. You will find that there are 6 possible configuration for those blocks.
For every configuration, look at couple of exponents $(\alpha,\beta)$ of the fator $(x-\lambda)$ in $\chi$ and $\mu$. You vill find that the couples $(\alpha,\beta)$ are distinct in each of the 6 configurations, so $\chi$ and $\mu$ are enough to uniquely determine the Jordan form.