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Given two 6x6 nilpotent matrices with the same minimal polynomial and same rank, prove they're similar. Also prove that this is not necessarily the case if the two matrices are 7x7.


If two matrix have the minimal polynomial and same rank, then the following can be generalized:

1) they have the same eigenvalue, 0

2) then have the same nilpotent index

3) they have the same geometric multiplicity

But I'm not seeing how this can explicitly imply similarity and how the 7x7 case is any different.

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Note that two matrices are similar if and only if they have the same Jordan canonical form. With that in mind, note that the Jordan forms of the matrices we're discussing:

  • Must have only $0$-blocks (they only have $0$ as an eigenvalue)
  • Must have the same total number of blocks (since they have the same rank)
  • Must have the same maximal block-size (since they have the same minimal polynomial)

In the $6 \times 6$ case, this is enough to ensure that the Jordan forms are the same (up to a permutation of the blocks). In the $7 \times 7$ case, we have counterexamples such as the pair $$ \pmatrix{ 0&1\\&0&1\\&&0\\ &&&0&1\\ &&&&0&1\\ &&&&&0\\ &&&&&&0}, \quad \pmatrix{ 0&1\\&0&1\\&&0\\ &&&0&1\\ &&&&0&\\ &&&&&0&1\\ &&&&&&0} $$

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