Questions tagged [nilpotence]

A nilpotent element of a ring has $a^n=0$ for some integer $n$.

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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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Sufficient condition for nipotency of a Lie group

Let $G$ be a Lie group such that its Lie algebra $\mathfrak{g}$ admits a decomposition of the form $$\mathfrak{g}=\Delta\oplus [\mathfrak{g},\mathfrak{g}].$$ Where $\Delta$ is a bracket generating ...
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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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If $\operatorname{char}R=p$ and $a\in R$ is nilpotent, then $1+a$ has finite order in $U(R)$ [duplicate]

Let $R$ be a commutative unit ring such that $\operatorname{char} R=p$ ($p$ is a prime) and let $a\in R$ be a nilpotent element, $a^n=0$. We know that $1+a$ is a unit in $R$. Prove that $1+a$ has ...
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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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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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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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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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Non-nilpotent Elements Not Contained in Every Prime Ideal, But What About All Maximal Ideals [duplicate]

I am currently studying for an algebra qualifying exam and am stuck on the second part of a problem. Assume R is a commutative ring (it doesn't specify if the ring has a multiplicative identity). The ...
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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 ...
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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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How to prove potencies of matrices by induction

a) The matrix has the property ajk = 0 for 1≤j≤k≤N. Proof: A^N = 0. Note: use the induction over N. b) With the help of a), calculate the expression B^20 for the real matrix enter image description ...
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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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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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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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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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$ 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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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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How can a positive factor change the sign of an expression? Need intuitive explanation.

From this answer it follows that $\operatorname{sign} X \ne \operatorname{sign} a X$ for any $a>0$ and nilpotent $X$. For instance, in dual numbers, $\operatorname{sign} \varepsilon \ne \...
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$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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Norm of complex symmetric nilpotent matrix

Let $A$ be a complex square matrix. We know that $A$ is symmetric but not Hermitian: $A = A^T$ and $A \neq A^*$. $A$ is nilpotent. In fact, we have $A^2 = 0$. I need to bound the norm of $A$ in any ...
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Linear algebra nilpotent linear transformation

I was reading canonical form: Nilpotent linear transformations of linear transformation from Topics in Algebra I.N. Herstein. I have one doubt in proof of lemma 6.5.4 statement of lemma is as follows: ...
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Nilpotent Operators over Vector Spaces

Exercise Prove that if $S:V \to V$ (a vector space) is nilpotent, then $I_V-S$ in invertible. An operator $S: V \to V$ is nilpotent if there some some natural number $k$ such that $S^k = \underbrace{...
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A nilpotent element in $\mathbb{Z}/ 96\mathbb{Z}$

I have to find a nonzero nilpotent element in the ring $\mathbb{Z}/ 96\mathbb{Z}$. If we take $a\in \mathbb{Z}/ 96\mathbb{Z}$, $a$ is nilpotent if there exists $n\in\mathbb{Z}$ such that $a^n = 0$. I ...
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Why is A not 1 nilpotent?

I am trying to read a solution for this question: Given the following matrix in Jordan form, $$ A= \begin{pmatrix} J_5(0) & \\ & J_6(0) \end{pmatrix} $$ We are asked to ...
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Hoffman Kunze Section 6.8 Exercise 12

In the Linear Algebra book by Hoffman Kunze, Exercise 12 of section 6.8 goes If you thought about Exercise 11, think about it again, after you observe what Theorem 7 tells you about the ...
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Lower central series of the Unitriangular group $UT(n, \mathbb{Z}_p)$

This is Exercise 5.44 from Rotman's book "An Introduction to the theory of Groups (4th Ed)". Specificaly, the exercise asks us to prove that the $i$-eth term in the lower central series is ...
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Proof of Nilpotency of Matrix All Whose Powers Have Vanishing Trace Without Using Eigenvalues

Let $k$ be a field and let $A\in M^{n\times n}(k)$ be such that $tr(A^m)=0$ for all $m\in \mathbb N$ (or all $m\leq n$). Prove that $A$ is nilpotent. The canonical proof can be found in this answer. ...
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Let $A \in M_n(C)$, $A^m = 0 \implies A^n = 0$

Consider a matrix $A$ of size $n$ over $C$. I tried to show that if $A^m =0$ for some $m$ then $A^n =0$ We may assume that WLOG there exist a vector $x \ne 0$ such that $A^{m-1} x \ne 0$. Suppose $...
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What does it mean that lower central series terminate?

The definition of nilpotent Lie algebra $\mathfrak{g}$ says that its lower central series terminate at the zero subalgebra. The lower central series is the $\mathfrak{g}$, then scalar product of $\...
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Which of the following options must be true.

Let $A$ be a $2×2$ non zero matrix with entries in$\Bbb C$ such that $A^2 = 0$. Which of the following statements must be true ? $PAP^{-1}$ is diagonal for some invertible $2×2$ matrix $P$ with ...
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Let $A_{n×n}$ be a complex matrix then prove that $ch_{AB}(x) = ch_{BA}(x)$

Let $A_{n×n}$ be a complex matrix then prove that $ch_{AB}(x) = ch_{BA}(x)$ I saw somewhere that $ch_{AB}(x) = ch_{BA}(x)$ if $A$ is invertible and if $AB$ is nilpotent then $BA$ is nilpotent. What ...
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An association between finite nilpotent groups and Lie rings

The following line turns up in a paper without much of a reference: Let $G$ be a nilpotent groups of nilpotency class $c$, and suppose $V$ be its associated Lie ring, where $V=V_1\oplus V_2\cdots \...
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Nilpotent elements of $R$: If $x \in \operatorname{Nil}(R)$ then $1+x \notin P$, where $P$ is any prime ideal of $R$.

Denote $\operatorname{Nil}(R)$ as $N$. We already have than $N$ is an ideal in $R$. It remains to prove that: $N \subseteq P$ for any prime ideal, $P$, in $R$ and then conclude that $\forall x \in N,...
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How to construct nilpotent of order $3$

Let $A = \pmatrix {a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}}$ Let $A \neq 0$ then construct a nilpotent matrix such that (1) $A^2 = 0$ (...
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$C^m=0$ for some $m$, $AC=CA$, can we show $A+C,A$ has the same eigenvalues? [duplicate]

$C^m=0$ for some $m$, $AC=CA$, can we show $A+C,A$ has the same eigenvalues? I came out this problem when I treat the following problem. Let $A,B,C$ be three $n\times n$ comple matrices, $AB-BA=C$, $...
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If $A\in M_n(\mathbb{C})$ is a nilpotent matrix then $A$ is similar to $2A$

If $A\in M_n(\mathbb{C})$ is a nilpotent matrix then $A$ is similar to $2A$ I am trying to prove this property but the truth is I cannot find how to express the matrix $ P $ such that $$ A=P^{-1}2AP \...
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Why do successive compositions of nilpotent operators strictly decrease their rank?

$\DeclareMathOperator{\N}{\mathcal{N}}\DeclareMathOperator{\Ker}{Ker}\DeclareMathOperator{\id}{Id}\DeclareMathOperator{\rk}{Rank}$It is important in my reading of a formal proof of the existence of ...
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$\mathfrak{Jac}(A[x])=\mathfrak{Nil}(A[x])$ [duplicate]

Let $A$ be a commutative associative ring. Show that $\mathfrak{Nil}(A[x])=\mathfrak{Jac}(A[x])$. Since $\mathfrak{Nil}(A[x])=\bigcap\limits_{\mathfrak p\triangleleft A[x]}\mathfrak{p}$, it's ...
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Does every nilpotent element in a real Lie algebra lie in some minimal parabolic Lie algebra?

Definitions: Let $\mathfrak{g}$ be a finite-dimensional, real, semisimple Lie algebra with some Cartan involution $\theta$. Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ be the usual Cartan ...
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Does a nilpotent operator always have a Jordan representation?

(Supposing the Jordan representation exists) Since a Jordan block associated to $\lambda$ has the form $J= \lambda I + N$, where $I$ is the identity matrix and $N$ a nilpotent matrix. One could find a ...
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Help with Nilpotent Transformation Proof

I have the following linear algebra textbook question and I'm not really sure where to begin with it: Let $V$ be a (possibly infinite-dimensional) vector space. We say a linear map $T: V \rightarrow V$...
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Proving that $\det(A+N) = 0$ with N a nilpotent matrix and A a singular matrix such that AN=NA

Let $N \in \mathcal{M}_{n}(\mathbb{C})$ be a nilpotent matrix of indice $m$ and $A \in \mathcal{M}_{n}(\mathbb{C})$ a matrix such that $A N=N A$. We assume that $A$ is singular. Expressing $(A+N)^{k}$ ...
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Find $\{X^{T}AX:X \in\mathbb{R}^n\}$ where $A$ is nilpotent.

Find $S = \{X^{T}AX:X \in\mathbb{R}^n\}$ where $A \in \mathcal{M}_{n,n}(\mathbb{R})$ is nilpotent. What I have done so far: If $A$ is nilpotent then its only eigenvalue is $0$ and $A^n = 0$. If $A = ...
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Prove that if ${\frak g}\le{\frak gl}(V)$ and $\mathfrak h\le\mathfrak g$ is a maximal subalgebra, then $\mathfrak h$ is an ideal of codimension $1$

Let ${\frak g}\le{\frak gl}(V)$ be a finite-dimensional subalgebra of nilpotent endomorphisms on some vector space $V$, and let ${\frak h}\le{\frak g}$ be a maximal subalgebra of $\mathfrak g$. Using ...
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$A \in GL_n (\mathbb C)$ and $N$ nilpotent matrix, A and N commute. Show there exists B such as $B^{2} = A + N$

Let $n \in \mathbb N^{*}$, $A \in GL_n (\mathbb C)$ and $N \in M_n (\mathbb C)$ with $N$ Nilpotent such as $AN=NA$. Prove that there exists $B \in M_n (\mathbb C) $ such as $B^{2} = A + N$ Since $A$ ...
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A problem nilpotent linear operator

Let T be a nilpotent linear operator on the vector space $\mathbb{R}^5$. Let $d_i$ denote the dimension of the kernel $T^i$.Which of the following can possibly occur as a value of $(d_1,d_2,d_3)$? (a)$...
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