Simultaneously Strictly Triangularizable

Algebra Level 5

If A A and B B are nilpotent matrices of the same order over C \mathbb C , which of the following statements is/are true?

Ⅰ. If A A and B B are simultaneously strictly triangularizable, that is, there exists a similarity matrix P P such that P 1 A P P^{-1}AP and P 1 B P P^{-1}BP are both strictly upper triangular, then A A and B B commute.

Ⅱ. If A A and B B commute, then they can be simultaneously strictly triangularizable.


Bonus: Generalize this for any number of matrices.

Neither Ⅱ only Ⅰ and Ⅱ Ⅰ only

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1 solution

Patrick Corn
May 22, 2019

I is clearly false: for instance A = ( 0 1 0 0 0 0 0 0 0 ) , A = \begin{pmatrix} 0&1&0 \\ 0&0&0 \\ 0&0&0 \end{pmatrix}, B = ( 0 0 0 0 0 1 0 0 0 ) . B = \begin{pmatrix} 0&0&0 \\ 0&0&1 \\ 0&0&0 \end{pmatrix}.

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).

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