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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.

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    – user1551
    Oct 22, 2021 at 10:21

1 Answer 1

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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

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

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