If $A,B \in \mathbb{R^n}$ are two diagnolizeable matrices with same characteristics polynomial then $A$ and $B$ are similar.
I tried to find a solution whether this is true or not.
I think this is true because if both matrices are diagnolizeable then they both have basis consisting of $n$ eigenvectors and since characteristics polynomial is also same so there eigenvalues must also be same $\rightarrow Tr(A) == Tr(B)$ and $det(A) == det(B)$.
The other way i tried is: we can write both $A$ and $B$ as diagonal matrix,such that both $A$ and $B$ only have eigen values as diagonal elements and all other entries should be zero but here i don't know whether every diagonal matrix is similar to everyother diagonal matrix, I could not find any counter example nor i know any proof?
I don't even know whether the lemma is true or not but the process above lead me to yes. Can someone tell me if this is true and how to prove it?