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### Kernel of AB if $[A,B]=0$ and $AB\neq0$? [closed]

I have found similar results here and mathematics stack exchange but they all imposed specific conditions that don't suit this problem in particular. The problem is as follows.
Let A,B be square ...

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### A question on Nilpotent Matrix

Suppose we have a linear matrix space $S\subset M_{n\times n}$, any $M\in S$ is a nilpotent matrix,
that is $M^n=0$.
Then for any finite subset of $S$, says $A=${$M_1,...,M_k$}, one can define the ...

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### When is $\ker AB = \ker A + \ker B$?

Prove/ Disprove: Let $n$ be a positive integer. Let $A$, $B$ be two $n \times n$ square matrices over the complex numbers. If $AB = BA$ and $\ker A = \ker A^2$ and $\ker B = \ker B^2$
then $\ker AB = ...

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### Conjugacy class of a full Jordan block over integers

Can we characterize all integer matrices that are similar (over $\Bbb Z$) to a full Jordan block with $0$'a on the diagonal? In other words, can we determine the conjugacy class of such a matrix over ...

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### Terminology for nilpotent groups

I have a nilpotent lie group $N$ with upper central series
$$1 = N_0 \triangleleft N_1 \triangleleft \dots \triangleleft N_k = N$$
which induces the filtration $$0 = \mathfrak{n}_0 \subset ...

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### Request for info on the space of commuting matrices preserving a flag.

Fix a flag of subspaces V1 in V2 in V3, etc. all in Cn.
Consider the space of pairs of commuting linear transformations A and B such that:
A preserves the flag (i.e. A(Vi) is in Vi), and
B strictly ...