Questions tagged [nilpotent-matrices]
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18 questions
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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 = ...
- 685
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 ...
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2
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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\...
- 673
3
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3
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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 $...
- 545
3
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2
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355
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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}...
- 4,014
3
votes
1
answer
145
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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 ...
3
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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 ...
- 6,562
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).
...
- 1,677
2
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1
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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\}...
user76479
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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 ...
- 24.8k
1
vote
1
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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 ...
- 1,503
1
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1
answer
321
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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 ...
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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 ...
- 769
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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 ...
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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\...
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0
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1
answer
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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 ...
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0
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0
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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 ...
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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.
...
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