# Questions tagged [nilpotent-matrices]

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18
questions

3
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### The rank of a certain linear combination of mutually commuting nilpotent matrices

Let $A_1,\ldots,A_r$ be mutually commuting $n\times n$ nilpotent matrices over $\mathbb C$, the field of complex numbers. For any complex number $c$, let $A(c):=A_0+cA_1+c^2A_2+\ldots +c^rA_r$. We ...

0
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0
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241
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### Simultaneous triangulation and Jordan normal form of commuting nilpotent matrices

Let $A_1,\ldots,A_r$ be $n\times n$ nilpotent matrices over $\mathbb C$, the field of complex numbers, satisfying $A_i\cdot A_j=A_j\cdot A_i$ for all $i,j$. As the matrices commute, they admit ...

1
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0
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### About nilpotent Jordan algebras, matrix representations and formally real algebras

Given an non-commutative associative unital algebra A of characteristic $0$, one can construct a Jordan algebra $A+$ using the same underlying addition vector space.
Notice first that an associative ...

0
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0
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131
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### On nilpotent singular $\mathbb F_2^{n\times n}$ matrices

Let $M$ be a $0/1$ matrix over $\mathbb F_2^{n\times n}$ with determinant $0$.
The set of such singular matrices form a semigroup.
The set of nilpotent matrices of size $n\times n$ form a semigroup.
...

5
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2
answers

395
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### Can the concatenation of projection operators be nilpotent with an index k>=3?

Let $\boldsymbol{V}_{1},\dots,\boldsymbol{V}_{n}\in\mathbb{R}^{d\times m}$ be $n$ “tall” matrices (where $d\ge m$) with orthonormal columns.
And let $\boldsymbol{P}_{1},\dots,\boldsymbol{P}_{n}\in\...

2
votes

0
answers

253
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### For a nilpotent matrix A, are the cardinalities of sets: 1) B: commute with A, 2) B: anticommute with A, 3) B: q-commute with A — the same?

Let us work over finite fields $F_{p^k}$. Simulations seems to indicate:
Question 1: Consider a nilpotent matrix $A$, consider the set of all matrices $B$, such that $AB-qBA=0$, then cardinality of ...

1
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0
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120
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### Algorithm for the nilpotence of matrix polynomials

Let $P$ be a multivariate polynomial of real-valued $N \times N$ matrices. Given $X_1, X_2, ..., X_M \in \mathcal{M}_N\{\mathbb{R}\}$, is there any optimal algorithm to determine whether the result of ...

3
votes

0
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109
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### cohomology of nilpotent matrices of fixed $m$-th power

Let $k$ be an algebraically closed field, $\mathcal{N}$ is the variety of $n \times n$ nilpotent matrices over $k$, and consider the natural $m$-power map $\mathcal{N} \rightarrow \mathcal{N}$ given ...

2
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2
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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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1
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124
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### 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

1
answer

356
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### 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

2
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833
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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 $n\...

1
vote

1
answer

376
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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 ...

20
votes

3
answers

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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 = ...

3
votes

3
answers

728
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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 $...

6
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4
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### 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

2
answers

351
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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 \mathfrak{n}...

1
vote

1
answer

316
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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 ...