Non-conjugate matrices having the same minimal & characteristic polynomials & the same dimension of the eigenspace 
What is the smallest value of $n$ for which there are two non-conjugate $n\times n$ matrices which have the same minimal and characteristic polynomials and eigenspaces of equal dimension?

I first thought that the answer would be 4, as we know that for $n<4$ having the same minimal and characteristic polynomial implies similarity, but since we have the additional condition that they must have eigenspaces of equal dimension, I am not sure.
 A: Let $k$ be an algebraically closed field.
Facts: Let $M$ be a matrix with coefficients in $k$. Then $M$ is conjugate to a block-diagonal matrix where each block is a Jordan block. These blocks, up to reordering, classify the conjugacy classes.
Moreover, let $\lambda$ be an eigenvalue of $M$.
Then

*

*the number of Jordan blocks corresponding to $\lambda$ is the dimension of the eigenspace;

*the size of the largest block corresponding to $\lambda$ is the multiplicity of $\lambda$ in the minimal polynomial;

*the sum of the sizes of the blocks corresponding to $\lambda$ is the multiplicity of $\lambda$ in the characteristic polynomial.

Therefore, it is enough to look for the smallest integer $N$ such that there are an integer $k$, and two different $k$-tuples of integers $(n_1,\cdots,n_k)$ and $(m_1,\cdots,m_k)$ such that:

*

*$n_1\geq \cdots \geq n_k \geq 1$ and $m_1\geq \cdots \geq m_k \geq 1$;

*$\sum^k_{i=1} n_i = \sum^k_{i=1} m_i = N$;

*$n_1 = m_1$;

and to form two matrices with only one eigenvalue having Jordan blocks of sizes $(n_1,\cdots,n_k)$ and $(m_1,\cdots,m_k)$.
It is easily seen that the smallest $N$ is $7$, and that examples of tuples are $(3,3,1)$ and $(3,2,2)$.
An example of a couple of two matrices is the following:
$
\begin{pmatrix}
1&1&0&0&0&0&0\\
0&1&1&0&0&0&0\\
0&0&1&0&0&0&0\\
0&0&0&1&1&0&0\\
0&0&0&0&1&1&0\\
0&0&0&0&0&1&0\\
0&0&0&0&0&0&1\\
\end{pmatrix}$ and
$\begin{pmatrix}
1&1&0&0&0&0&0\\
0&1&1&0&0&0&0\\
0&0&1&0&0&0&0\\
0&0&0&1&1&0&0\\
0&0&0&0&1&0&0\\
0&0&0&0&0&1&1\\
0&0&0&0&0&0&1\\
\end{pmatrix}.$
