# 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.

• $7.\phantom{ }$ Oct 22, 2021 at 10:21

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

$$\begin{pmatrix} 1&1&0&0&0&0&0\\ 0&1&1&0&0&0&0\\ 0&0&1&0&0&0&0\\ 0&0&0&1&1&0&0\\ 0&0&0&0&1&0&0\\ 0&0&0&0&0&1&1\\ 0&0&0&0&0&0&1\\ \end{pmatrix}.$$