Questions tagged [nilpotent-matrices]

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Simultaneous upper-triangularization for two nilpotent commuting matrix

Given two nilpotent matrix B1 and B2 over complex numbers which commute i.e. [B1,B2]=0, we know that they can be conjugated to upper-triangular ones (even strictly-triangular since they're nilpotent). ...
0
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1answer
73 views

About indecomposability and nilpotence

Transferred from MSE where it now received a complete answer. Maybe the following is easy, but I am not an expert in finite-dimensional Lie algebras and was stuck on the following problem. Can ...
2
votes
1answer
241 views

Commuting nilpotent matrix collection

For every large enough $m\in\Bbb N$ are there $c=\alpha m$ (for some fixed $\alpha>0$) square matrices $A_1,\dots,A_c$ that commute with each other with nonzero product ($\forall i,j\in\{1,\dots,t\}...
0
votes
2answers
639 views

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 $n\...
1
vote
1answer
312 views

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 ...
20
votes
3answers
3k views

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 = ...
3
votes
3answers
610 views

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 $...
5
votes
4answers
1k views

The eigenvalues of the sum of two nilpotent matrices

I have a matrix that is given by $A e^{i q} + A^* e^{-i q}$ with $A$ a nilpotent $n\times n$ matrix. The eigenvalues I get turn out always to be independent of $q$ but I cannot prove it. I want to ...
3
votes
2answers
340 views

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 \mathfrak{n}...
1
vote
1answer
259 views

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 ...