Let $A$ and $B$ be $n \times n$ real matrices with same minimal polynomial. Then
(i) $A$ is similar to $B$.
(ii) $A-B$ is singular.
(iii) $A$ is diagonalizable if $B$ is so.
(iv) $A$ and $B$ commute.
I think only (iii) is the correct option, similar matrcies have same characteristics polynomial but the converse may not be true, $\begin{pmatrix}1&0\\0&0\end{pmatrix}\times \begin{pmatrix}0&1\\0&1\end{pmatrix}\ne \begin{pmatrix}0&1\\0&1\end{pmatrix}\times \begin{pmatrix}1&0\\0&0\end{pmatrix}$ though they have same minpoly $x(x-1)$ the same two matrices works as a counter example for (ii), am I right ?
Best Answer
You are correct, and here are the two missing counterexamples:
(i) $\begin{pmatrix}1&1\\&1\\&&1\\&&&1\end{pmatrix}$ and $\begin{pmatrix}1&1\\&1\\&&1&1\\&&&1\end{pmatrix}$ both have minimal polynomial $(x-1)^2$, but are not similar. (For example, similar matrices have the same dimension of the corresponding eigenspaces.)
(ii) Take $A=\begin{pmatrix}1\\&2\end{pmatrix}$ and $B=\begin{pmatrix}2\\&1\end{pmatrix}$, then $A-B=\begin{pmatrix}-1\\&1\end{pmatrix}$ is non-singular.
(iii) DonAntonio provided an excellent explanation why this is true in this comment.
(iv) Your counterexample, that this is not true, is correct.