# Example of matrices with some interesting properties like same characteristic and minimal polynomial etc.

Looking for two matrices $A$ and $B$ with entries in the field $F_2$ with the following properties:

$A$ and $B$ both are invertible,have same minimal polynomial,Characteristic polynomial,same dimension of each eigenspace and both are not conjugate to each other.

I can think of matrices which have some properties like have same characteristic polynomial and minimal polynomial but I am unable in finding example with all above proprties.Help!

Since the questions asks about similar matrices, the first idea is to think about matrices $A$ and $B$ in Jordan form with exactly one eigenvalue. Then the characteristic polynomials will agree. Since dimension of eigenspace is equal the number of Jordan blocks, $A$ and $B$ should have same number of Jordan blocks. Since $A$ and $B$ should have same minimal polynomial, the largest Jordan blocks of $A$ and $B$ have to be equal.
It turns out see this question that the minimal size, where this is possible, is $7$.
Take $$A = \pmatrix{ 1&1&1&&&&\\ &1&1&&&&\\ &&1&&&&\\ &&&1&1&1&\\ &&&&1&1&\\ &&&&&1&\\ &&&&&&1 }, \quad B = \pmatrix{ 1&1&1&&&&\\ &1&1&&&&\\ &&1&&&&\\ &&&1&1&&\\ &&&&1&&\\ &&&&&1&1\\ &&&&&&1 }$$ Their minimal polyniomial is $(t-1)^3$, the characteristic polynomial is $(t-1)^7$, the dimension of the eigenspace to eigenvalue $1$ is $3$. They are not similar (or conjugate), as $$5 = \dim\ker((I-A)^2)\ne\dim\ker((I-B)^2)=6.$$