If and are nilpotent matrices of the same order over , which of the following statements is/are true?
Ⅰ. If and are simultaneously strictly triangularizable, that is, there exists a similarity matrix such that and are both strictly upper triangular, then and commute.
Ⅱ. If and commute, then they can be simultaneously strictly triangularizable.
Bonus: Generalize this for any number of matrices.
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I is clearly false: for instance A = ⎝ ⎛ 0 0 0 1 0 0 0 0 0 ⎠ ⎞ , B = ⎝ ⎛ 0 0 0 0 0 0 0 1 0 ⎠ ⎞ .
A general form of II is proved by Theorem 6.4 in this expository paper (note that for a nilpotent matrix, any upper triangularization must be a strict upper triangularization, because the eigenvalues are the diagonal entries, and they must all be zero).