$A,B\in M_{n\times n}(K)$ diagonalizable with equal characteristic polynomials. Are they similar?

Question

Let $A,B\in M_{n\times n}(K)$ be diagonalizable over $K$. Additionally, let their characteristic polynomials be equal: $p_A(\lambda)=p_B(\lambda)$. Does this imply $A$ and $B$ are similar? I am able to show the implication in the opposite direction: if $A$ and $B$ are similar, then they do have the same characteristic polynomials. Do you have any suggestions?

Answer 1

If they have the same characteristic polynomials, then they have the same eigenvalues with the same (algebraic) multiplicities, so their diagonal forms must be the same.

And if $A$ and $B$ are similar to the same diagonal matrix, then ipso facto they are similar to each other -- similarity is transitive.

The assumption that $A$ and $B$ are both diagonalizable is crucial. For example, $({}^2_0\,{}^1_2)$ and $({}^2_0\,{}^0_2)$ have the same characteristic polynomial, but are not similar.