Questions tagged [nilpotence]
A nilpotent element of a ring has $a^n=0$ for some integer $n$.
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Which Lie algebras are derived algebras?
Let $\mathfrak{h}$ be a Lie algebra over a zero characteristic algebraically closed field.
When is $\mathfrak{h}$ the derived algebra of some Lie algebra $\mathfrak{g}$?
This clearly imposes some ...
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votes
1
answer
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Subspace of square matrices where all matrices are nilpotent and the products of the matrices are also in the subspace.
Let $A$ and $B$ be $3\times3$ square matrices (maybe could also examine $n\times n$) and let $U$ be a subspace where $A,B,AB\in U$. All matrices in $U$ are nilpotent. I conjecture that all matrices in ...
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Proving if $I$ nilpotent and $I\neq N(R)$ then $R/I$ has nilpotents
I am trying to prove that if $R$ is a ring and $I$ is a nilpotent ideal of $R$ and $I\neq N(R)$ then $R/I$ has a nonzero nilpotent element.
My attempt
Since $I$ nilpotent ideal of $R$ we have $I \...
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Computing the nilradical of a ring
Let $R=\begin{pmatrix}
\mathbb{C} & \mathbb{C} & \mathbb{C}\\
0 & \mathbb{C} & \mathbb{C}\\
0 & \mathbb{C} & \mathbb{C}
\end{pmatrix}$. I want to find the nilradical: $$N(R)=\...
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Square root from nilpotents
We know that
$$\left(
\begin{array}{cccc}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
\end{array}
\right)^2=\left(
\begin{...
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Fulton & Harris Proof Clarification for Lemma D.6
Lemma D.6 of Fulton & Harris claims
If $H$ is regular, then $\mathfrak g_0(H)$ is abelian,
where $\mathfrak g$ is a complex Lie algebra, $\mathfrak g_0(H)=\ker\operatorname{ad}(H)^m$ for large $...
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Matrix multiplication and nilpotent matrix
Given that you have an $n\times n$-matrix called $A$ with the property that $A^k = 0$. Prove that $A^n = 0$.
Have heard about eigenvalues and null space as ways for the proof. The thing is that I can'...
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votes
1
answer
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For which $n \in \mathbb N$ does $A = \mathbb Z/ n \mathbb Z$ has $\operatorname{Nil}(A) \neq 0$?
I want to try to answer the following question:
For which $n \in \mathbb N$ does $A = \mathbb Z/ n \mathbb Z$ has $\operatorname{Nil}(A) \neq 0$?
Here are my thoughts:
I know that $\operatorname{Nil}(...
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Is there a name for the sum of a scalar + nilpotent operator?
Let $V$ be a vector space and suppose $T\in\operatorname{End}(V)$ is the sum of a scalar operator and a nilpotent operator. That is, $T=T_s+T_n$, where $T_s=\lambda I$ for some scalar $\lambda$, and $...
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Nilpotent elements in $\mathbb{R}[x]/\langle x^4+4\rangle$
Let $\mathbb{R}[x]$ denote the ring of polynomials in $X$ with real coefficients. Then, the quotient ring $R=\mathbb{R}[x]/(x^4+4)$ is
(a) field
(b) an integral domain, but not a field
(c) not an ...
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votes
2
answers
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Nilpotent Matrix And Sequence Properties
The original problem from algebra book asks to prove that if $A$ is a $2\times2$ nilpotent matix then $A^2=0$ . Can this be related with some properties of exact sequences ? I.m. that such matrix is a ...
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1
answer
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Is it ok to have a nilpotent matrix with index $0$, and is there a short-cut to find the index?
I am thinking about the definition of a nilpotent matrix, what I know is that a nilpotent matrix is a square matrix, say matrix $A$, such that $A^k$ is a null matrix with index $k$.
According to this, ...
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vote
1
answer
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How to find in practise a $\mathfrak{sl}_2$-triple in a simple lie algebra?
I want to justify the existence of a $\mathfrak{sl}_2$-triple in a simple lie algebra.
I know there exists the Jacobson-Morozov theorem. It states that given a nilpotent element of the lie algebra, ...
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votes
1
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Nilradical in an Artinian ring
I am reading a book that suggests to find what $N(R)$ is - the nilradical of $R$ where $R$ is Artinian you just need to find a nilpotent ideal $I $ of $R$ and then show that $R/I $ has no nonzero ...
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Nilpotent ideals in a ring proof
I am trying to understand a proof I am reading but it doesn’t make much sense to me.
If $R$ is a ring with $I$ a nilpotent left ideal then $I$ is contained in a nilpotent two-sided ideal.
The proof ...
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1
answer
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Is $N_T \subseteq Im(T^l)$?
If $T\in \mathcal L(V)$ is a nilpotent linear map (ie. $T^k=0_V$ for some $k \in \mathbb{N}_{\ne 0}$), let $N_T$ be the nullspace of $T$, $Im(T^l)$ be the image of $V$ under $T^l$.
Question: Is $N_T \...
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When is a family of nilpotent matrices over $\mathbb{F}_p$ zero?
Given some $n\times n$ matrices $ A_{i},i=1,\dots,m $, over a finite field $\mathbb{F}_p$. Suppose that all $ A_{i} $ are nilpotent and
$$ c_{i}A_{i}+\sum_{j\neq i}b_{ij}[A_{i},A_{j}]=0 $$
for some $ ...
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0
answers
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show that $AB-BA$ is nilpotent [duplicate]
let the set of all n by n matrices with complex entries be denoted by $M_n(\mathbb{C}).$ Let $A,B\in M_n(\mathbb{C})$ be such that $AB-BA $ commutes with $A$. Show that $AB-BA$ is nilpotent.
I need ...
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Ideals generated by all nilpotent elements in noncommutative rings
It is a basic fact that a set of all nilpotent elements in a commutative ring is an ideal. Suppose that $A$ is a noncommutative ring, $I$ is a two-sided ideal generated by all nilpotent elements and $...
1
vote
1
answer
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views
Let $R=\mathbb Z_{36}$. Find $N(\langle 0 \rangle), N(\langle 4\rangle), N(\langle 6 \rangle)$.
Let $R=\mathbb Z_{36}$. Find $N(\langle 0 \rangle), N(\langle 4\rangle), N(\langle 6 \rangle)$.
Let $R$ be a commutative ring and let $A$ be any ideal of $R$. Then the nilradical of A, $N(A)=\{r\in R:...
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3
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Invertible element can't be nilpotent?
When I was reviewing cracking GRE subject mathematics 4-th edition, I was confused about the proof in Page 247.
Consider invertible element c in a Ring, $cc^{-1} = 1$. Then for any integer n, $(cc^{-...
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Fitting subgroup of any group is locally nilpotent
Is it true that the Fitting subgroup of any group is locally nilpotent?
I know that if $G$ is finite then $F(G)$ is nilpotent and so locally nilpotent. Also, if $G$ is infinite then there's exists ...
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vote
0
answers
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Signature of endomorphism [closed]
I need some help on how to do this question please:
Let N ∈ M$_n$(F) be a nilpotent matrix and consider the endomorphism T induced on M$_n$(F) by the rule:
A → NA.
How is the signature of T related to ...
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votes
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answers
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$A$ is an $n$ by $n$ matrix over $\mathbb C$ such that $Rank(A)=1$ and $Tr(A)=0$, prove that $A^2=0$
This is what I did:
Because $\mbox{rank}(A)=1$, then from rank nullity theorem $$\dim\ker A + \mbox{rank} A = n \implies \dim \ker A = n-1$$
and $gm(0)=dimE(0)=dimKerA=n-1$ where $E(0)$ is the ...
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votes
2
answers
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Is every non-diagonalizable matrix diagonalizable over a larger non-reduced commutative ring?
Let $A$ be a square matrix with entries in some algebraically closed field $K$ (e.g., the complex numbers $\mathbb{C}$) and suppose that $A$ is not diagonalizable (so there is at least one eigenvalue ...
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votes
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Existence of maximal ideal in a commutative ring
Let $A$ be a commutative ring, $I \subsetneq A$ a proper ideal of $A$ and $a \in A$ such that $a^k \neq 0$ for all integer $k > 0$. Then there exists an ideal $J$ of $A$ that is maximal satisfying $...
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Rank of limit points of a conjugacy class
Problem: Let $t \to P_t$ be a one parameter subgroup $\mathbb{C}^* \to \text{Gl}_{n}(\mathbb{C})$.
Let $X$ be a $n \times n$ nilpotent matrix.
I want to show that if $lim_{t \to 0} P_t^{-1}XP_t = Y$ ...
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votes
0
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views
Prove an endomorphism is nilpotent [duplicate]
Let $V$ a finite dimensional vector space over a characteristic zero algebraically closed field. Let $x,y\in\mathfrak{gl}(V)$ and $[x,y]=z$ such that $z$ commutes with $x$ and $y$. Prove $z$ is a ...
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Is it true that nilpotent always has some eigenvalue?
I understand that if a nilpotent matrix has some $\lambda$ eigenvector, then it implies that $\lambda=0$ because if $$Ax = \lambda x \\ A^2x = \lambda^2x \\ A^3x=\lambda^3x \\ \vdots \\ 0=A^k=\lambda^...
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If $A$ has no non-trivial idempotents, then neither does $A/N$
Let $A$ be a commutative, associative, unital, finitely generated algebra over an algebraically closed field $k$. Denote by $N$ the nilradical of $A$, which is the set of all nilpotent elements of $A$ ...
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1
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Unipotent and semisimple elements are locally finite
Let $k$ be an arbitrary field, and let $V$ be an arbitrary $k$-vector space, possibly infinite-dimensional. Let $g\in\operatorname{End}_k(V)$. Then:
$g$ is diagonalizable if $V$ has a basis of ...
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votes
1
answer
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views
Constructing pairwise commuting nilpotent matrices
How can I construct $K$ mutually commuting nilpotent matrices $A_i$ with nilpotent index 3?
In other words, I need a set of matrices $A_i$ with following properties:
$A_i^3=0$ for $1\leq i \leq K$
$...
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1
vote
0
answers
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views
Does $AX+XB=B$ have a unique solution?
Let $A, B \in \Bbb C^{n \times n}$ such that $A$ is nilpotent but $B$ is non-nilpotent satisfying $AB+BA=O$. Then, $AX+XB=B$ has unique solution. True or false?
I don't understand how to approach ...
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Kernel and image of a nilpotent linear map
My question is as follows.
Let $T : V \to V$ be a linear map of a finite dimensional vector space V. If $ker T = Im T$, then is $dim V$ even and $T^{2} = 0$?
I believe this to be false since I have ...
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What is the index of nilpotency of $A$?
Here is the question I am trying to answer:
Find the Jordan form of
$$A = \begin{pmatrix}
1 & 0 & 1 \\
0 & 0 & 0\\
0 & 0 & -1
\end{pmatrix}$$
I know that the first step is to ...
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Prove if $T$ is nilpotent, $ST=TS$, and $\dim\operatorname{Nul}T=2$, then $S$ has at most $2$ distinct eigenvalues
Let $V$ be a finite-dimensional vector space, and let $S,T \in L(V)$ which commute, i.e.
$ST = TS$. Suppose $T$ is nilpotent with $\dim\operatorname{Nul}T = 2$. Prove that $S$ has at most $2$ distinct ...
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Constructing a strictly upper triangular nilpotent matrix
I asked previously about proving a lemma, where one can choose a basis for a linear operator such its matrix representation in this basis is strictly upper triangular. My original post is here:
How to ...
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How to show that for nilpotent linear operator $T$, there is a basis with respect to which the matrix of $T$ is strictly upper triangular [duplicate]
I need to prove the following:
Lemma: Suppose $T$ is nilpotent linear operator on $n$-dimensional complex vector space $V$. Then there is a basis of $V$
with respect to which the matrix of $T$ ...
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1
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if $ \dim \ker T^{k-1} < n , \dim \ker T^k = n $ and $ \dim V=n $ then $T$ is nilpotent
Statement: Let $ T: V \to V $ be a linear transformation over a finitely generated vector space $ V $. Suppose there exists $ k \leq n $ s.t. $\dim \ker T^{k-1} < n , \dim \ker T^k = n $ and $ \dim ...
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votes
2
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Is it possible for $AA^T$ to be a nilpotent matrix if neither $A$ nor $A^T$ are?
If $A$ is a square, non-nilpotent matrix with real-valued elements, and its transpose is $A^T$, then is it ever possible for $AA^T$ to be nilpotent? What if we allow complex-valued elements? Is $AA^H$ ...
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3
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Construct a $2022\times 2022$ matrix $A$ such that $A^{2022}=0$ but $A^{2021} \neq 0$
I want to construct a $2022\times 2022$ matrix $A$ such that $A^{2022}=0$ but $A^{2021} \neq 0$. More generally, I want to know how to construct a $n\times n$ matrix $A$ such that $A^n=0$ but $A^{n-1}...
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votes
3
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If a Lie algebra $\mathfrak{g}$ with only one Cartan subalgebra, then can we implies that $\mathfrak{g}$ is nilpotent?
We suppose that $\mathfrak{g}$ is a finite dimensional Lie algebra over $\mathbb{C}$. My professor told me that the assertion was clear, but I cannot figure it out. I know that each two Cartan ...
0
votes
1
answer
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$T \in \text{Hom}V $ is nilpotent implies $I - T$ invertible [duplicate]
Assume $T \in \text{Hom}V$ is nilpotent I.e. $T^n=0$ for some $n$. How do you show that this implies $I - T$ is invertible? The hint says the proof has something to do with the power series but I’m ...
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votes
1
answer
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views
If $N^n=0$ but $N^{n-1}\neq 0$, then there is no $n\times n$ matrix $A$ such that $A^2=N$
Let $n\in \mathbb N$ and let $N$ be an $n\times n$ matrix over the field $F$ such that $N^n=0$ but $N^{n-1}\neq 0$. Prove that there is no $n\times n$ matrix $A$ such that $A^2=N$.
I can understand ...
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0
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When does superdiagonal matrix have a single Jordan block
Let $A$ be the superdiagonal matrix given by $A=(a_{ij})$ where $a_{i,i+1}=r_i$ are nonzero entries and all the other entries are $0$. When is $A$ similar to a single Jordan block with eigenvalue $0$ ...
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vote
2
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Clarification regarding the number of Jordan blocks of some specific order
Good day! Last week at my algebra class I was asked to prove (and find) the following:
Given some nilpotent operator $T$ prove that for some number $k$, the number of Jordan normal blocks of order at ...
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votes
1
answer
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Linear algebra: nilpotent matrix and determinant.
My linear book had an exercise that demonstrated that a nilpotent matrix A has det(A)=0
$A^k=0$ is the nilpotent condition.
$det(A^k)=(det(A))^k$ and since $det(A^k)=0 \Leftrightarrow det(A)=0$.
My ...
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1
answer
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$ N$ is nilpotent and $ T $ is diagonalizable transformation s.t. $ N \circ T = T \circ N $. Show $ \ker q_{\lambda} ( T + N ) = V_{\lambda} $
Problem: Let $ V $ be finitely generated vector space over $ \mathbb{F} $, let $ T: V \to V $ be a diagonalizable linear transformation and $ N : V \to V $ a nilpotent linear transformation and ...
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1
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Let $ T: V \to V $ be nilpotent. Let $ f \in \mathbb{F[x]}. $ Show that $ f(T) $ is invertible iff $ f(0) \neq 0 $
Theorem: Let $ T: V \to V $ be nilpotent. Let $ f \in \mathbb{F[x]}. $ Show that $ f(T) $ is invertible iff $ f(0) \neq 0 $
Attempt:
$ ( \leftarrow ) $ Suppose that $ f(T) $ is invertible. Hence ...
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votes
0
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$R[x]$ is semiprimitive if and only if $R$ has no non-zero nilpotent ideals
Here is the question I'm working on: Show $R[x]$ is semiprimitive if and only if $R$ has no non-zero nilpotent ideals.
Here's my attempt at $semiprimitive \Rightarrow no\space non-zero\space ...
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