Questions tagged [nilpotence]

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

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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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Exercise on finding nilpotent elements

Task: Find the nilpotents in $\mathbb{Z}/\langle n\rangle$. In particular, take $n=12$. Solution: An element $m$ is nilpotent $\bmod{n}$ iff $n|m^k$. So $nil(R)=\{0,6\}$. My question about this ...
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Why Nilpotent Matrix is not Null matrix always?

I know it might sound dumb, but specifically, Why NULL matrices are not the only NULPOTENT matrices? I am thinking that as all eigen values of NILPOTENT matrices are 0, then $\lambda = 0$, and as per ...
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$tr(A)=tr(A^2)=tr(A^3) \implies A$ is Nilpotent [closed]

Let $A$ be a square matrix with $\mathrm{tr}(A)=\mathrm{tr}(A^2)=\mathrm{tr}(A^3)$. Prove that $A$ is nilpotent. The tip in the question is to find the matrix $B$ which is similar to $A$ and then $A^...
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Nilpotent orbits from either Dynkin diagram or Levi subalgebras

Given a nilpotent orbit, $\mathcal{O}_X$, associated to an element, $X$, of a complex semisimple Lie algebra, $\mathfrak{g}$, there are two equivalent classification schemes: Via the Jacobson-Morozov ...
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If $N$ is nilpotent, then $N \sim N^2 \Longleftrightarrow N=0$

Given that $N$ is nilpotent, then $N$ is similar to $N^2$ if and only if $N=0$ ($N$ is the zero matrix). The "$\Longleftarrow$" direction is easy enough, because $N = 0 = N^2$ so they're ...
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How to show this transformation is diagonalizable

Let $V$ be an finite dimensional vector space over $P$, $\mathscr{A}\in\operatorname{End}(V)$, $g(x)$ is a polynomial with $g(\mathscr{A})=O$, and $$ g(x)=\prod_{i=1}^m(x-\lambda_i)^{t_i} $$ Assume ...
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If $A^3 = O$, what condition do the entries of $A$ satisfy?

$$A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$$ When $A^3 = O$ (zero matrix) is satisfied, what condition do the four real numbers $a, b, c, d$ meet? I cubed $A$: $A^3 = \begin{...
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Show $φ^n= 0$ for $φ:V→V$ a linear endomorphism such that $\mathrm{tr}(\wedge^qφ) = 0$ and $\dim(V) = n$ a vector space. [duplicate]

As it is said in the title I'm working on an $n$-dimensional vector space $V$. My goal is to show that a linear endomorphism $\varphi:V \to V$ with $\mathrm{tr}(\wedge^qφ) = 0$ for all $q \geq 1$ is ...
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Prove that there exists a unitary matrix $U$ such that $U^H A U = B$

Given a non-null $3 \times 3$ matrix $A$ such that $A^2 = 0$, then prove that there exists a unitary matrix $U$ such that $$U^H A U = \begin{bmatrix} 0 & 0 & u \\ 0 & 0 & v \\ 0 & ...
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possible dimensions of a nilpotent endomorphism of a K-Vectorspace

I have the following exercise in linear algebra that I think I know what they're looking for, however have not really got an idea which theorems to use / how to approach this. Let $V$ be a 5-...
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Is there any property of the set of nilpotent matrices?

Let k be a field, $M_nk$ its matrix ring. We denote the set of nilpotent matrices by $N$. What properties can we know about $N$? Is there any analogy of "nilradical" in the matrix ring? ...
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Power associative basis implies not nilpotent?

Consider a non-associative commutative unital algebra of finite dimension where the product is defined by a cayley table such that elements are generated with real number coefficients $(a_0, \dots, ...
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$a_0 + a_1T + \cdots + a_k T^k$ is nilpotent if and only if $a_0 = 0$, where $T$ is nilpotent

Question: Let $T: V \rightarrow V$ be a nilpotent linear transformation of an F-vector space. Show that $H := \sum^k_{i=0} a_i T^i$ with $a_0, \ldots, a_k \in F$ is nilpotent if and only if $a_0 = 0$ ...
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Proving that the matrix representation of a nilpotent linear operator is upper-triangular with diagonal entries $ = 0$

Let T be a nilpotent operator on an $n$-dimensional vector space $V$, and suppose that $p$ is the smallest positive integer for which $T^P=T_0$ (zero-transformation). I have so far proved the ...
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Ring Theory + Combinatorics: the Multinomial Theorem

I'm working on the exact same problem as here and I'm curious about the converse $\Longleftarrow$ directions of (i) and (ii) Would it be possible to use the Multinomial Theorem to prove that if all ...
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All idempotents are central then $KG$ has no nilpotent

Given $G$ a finite group and $K$ a field of characteristic zero such that all idempotents in $KG$ are central, is it true that $KG$ has no nilpotent element or equivalently $KG$ has only division ring ...
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Given a rank-$1$ matrix $A$, what condition makes $A$ nilpotent?

My only idea on how to approach this problem was by making the matrix upper triangular. However, I found a simple counterexample to fact that elementary transitions do not change nilpotence of matrix. ...
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Every ideal $J \supsetneq \sqrt{0}$ contains a non-zero-divisor

I'm looking for an example of a commutative ring with unity, that has elements in each of the following three classes: Non-zero nilpotents Non-nilpotent zero-divisors Non-zero-divisors and ...
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Find all units, zero divisors and nilpotent elements of $\mathbb{Z}_4[X]$ and $\mathbb{Z}_6[X]$

I've been solving some problems from my abstract algebra course as training for the final exam, and I want to check if my solution to this one is correct: Describe the units, the nilpotent elements ...
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Example of a non-nilpotent matrix that gives $A^n(v)=0$ for some n

The question is as follows: Give an example of a non-nilpotent linear transformation for some vector space with the following property, for every $v \in V$ there is an $n$ for which $A^n(v)=0$. I ...
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A proof of the Jordan-Chevalley decomposition

Here is an exercise which is meant to prove the existence of the Jordan-Chevalley decomposition of a complex matrix. Let $A \in M_{n}(\mathbb{C})$, and let $\mu_{A} = \prod\limits_{k = 1}^{r} (X - \...
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Transforming of a nilpotent matrix to a specific form

I wish to build a question in the field of discrete state spaces representation (control theory). The canonical form has a very unique, but not singular, representation. I am focusing on the $A$ ...
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Invertibility of $I-A^s$ for each $s \geq 1$ implying nilpotence

If $R$ is a unital ring and $x \in R$ a nilpotent element, then $1-x$ is a unit, and so is $1-x^s$ for any $s \geq 1 $ since powers of $x$ are again nilpotent. My question concerns the converse of ...
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When a power of a prime ideal is not a primary Ideal

Take the ring $R=\mathbb{C}[x,y,z]/(x^3-y^3-z^3)$. I want to find a prime ideal, $p$, where $p^3$ is not a primary ideal. Any suggestions how to find such a prime ideal? I tried a few, like $(x,z)$, $(...
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Finding zero divisors of quotient rings $\mathbb{C}[x,y,z]/I$

I think (my maths could be wrong up to this point) that I am working with the ring: \begin{equation} \mathbb{C}[x, y, z]/(x^3 − y^3 − z^3,x^3,x^2z,xz^2,z^3)=\mathbb{C}[x, y, z]/(− y^3,x^3,x^2z,xz^2,z^...
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Atiyah-Macdonald: Exercise 3.5

Let $A$ be a ring. Suppose that, for each prime ideal $\mathfrak{p}$, the local ring $A_{\mathfrak{p}}$ has no nilpotent elements $\neq 0$. Show that $A$ has no nilpotent element $\neq 0$. If each $A_{...
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$A$ is nilpotent $4\times 4$ matrix and $\ker A^2 \ne \ker A^3$, Prove/Disprove the following

I'm stuck in this question and pretty lost here. Question : if $A$ is $4\times 4$ nilpotent matrix such that $\ker A^2 \ne \ker A^3$. Prove or disprove the following : a) Jordan form of $A$ has only ...
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127 views

Sum of nilpotent matrices with certain property is nilpotent

I want to prove that if $A$ and $B$ are nilpotent matrices such that $[A, [A,B]] = [B, [A,B]] = 0$ then $A + B$ is a nilpotent matrix as well. I know that it must follow from some general theorem from ...
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Error in my argument that $A^m=0$ implies $A^{m-1}=0$ for matrix $A$?

So nilpotent is the matrix where $A^m =0$ but $A^{m-1}$ is not equal to $0$. Well here's my logic- $$A^{m-1} = I \cdot A^{m-1} = A^{-1} \cdot A \cdot A^{m-1} = A^{-1} \cdot A^m = A^{-1} \cdot 0 = 0$$ ...
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Proving $3\times3$ nilpotent matrix satisfies $A^3=0$

Let $A$ be a $3$ $\times$ $3$ matrix. Prove that if there exists $n$ $>$ $3$ such that $A^n$= $0$, then $A^3$ = $0$. Can someone help explain how to prove this? I know that since A is nilpotent, ...
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Does strictly upper triangularity of $PAP^{-1}$ imply strictly upper triangularity of $PBP^{-1}$?

$A$ and $B$ are nilpotent matrices in $\mathbb M_n(F)$ that commute. Let $P$ be an invertible matrix such that $PAP^{-1}$ is strictly upper triangular. Does it also imply that $PBP^{-1}$ is also ...
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$A$ is nilpotent matrix and satisfies the equation $A^5=3A^3-2A.$ Prove $A=0.$

I'm stuck at this question , I had an idea how to solve it but I think it's not good tho. The Question : $A$ is nilpotent and $A^5=3A^3-2A,$ prove $A=0.$ ( $A^k = 0 :$ for certain $k\in \mathbb{N}$) ...
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Subset of nilpotent mappings of self maps is not open

Prove that the subset of all nilpotent mappings of $L(U)$ where $U$ is a finite dimensional normed vector space over $\mathbb{R}$ or $\mathbb{C}$ is not an open subset. Also, does this generalize to ...
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Calculating Nil(Z_n) faster

As far as I know, in order to calculate Nil(Z_n) I need to calculate each number raised to different values and see if I get 0. That is a very slow and inefficient way, in my opinion. Is there any ...
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Prove $0$ is the only eigenvalue of $A$.

Let $A$ be an $n$ x $n$ real matrix. Prove $0$ is eigen value of $A$. PROOF: Let $m∈ℤ^{+}$ so $A^{m}$x$=λ^{m}$x there exists$m≥1$ s.t $A^{m}=0$. $0=λ^{m}$x, but $x≠0$, so $λ^{m}=0$ $m>0$ , so $λ^{m}...
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Generalized eigenspace - Construction algorithm proof

Suppose that $A$ is an $n\times n$ nilpotent matrix. Let $x_1,\ldots,x_m$ be eigenvectors of $A$ forming a basis for the eigenspace of $A$. For $j=1,\ldots,m$, form a Jordan chain $$C\left(x_j\right)=\...
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Commutative ring: sum of nilpotent elements

I know that in a commutative ring, if $a$ and $b$ are nilpotent, then $a + b$ is nilpotent, and this can be proved using the binomial theorem. I cannot figure out, however, where the assumption of ...
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Killing form vanishes on nilpotent lie algebra

I come to you as a humble pilgrim of mathematics with yet another basic question... I thank you for your patience! So, lets say that $L$ is a nilpotent Lie algebra over field $F$. I'm trying to show ...
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$A$ is nilpotent, then $I+\lambda A$ is invertible for any $\lambda \in \mathbb{R}$

I need help in this problem Let $A$ is a square real matrix such that $A^{n}=0$ for some positive integer $n .$ Such a matrix is called nilpotent. Show that if $A$ is nilpotent, then $I+\lambda A$ is ...
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Would this be a valid proof for linear independence?

The question is a series of questions but this part gives an equation where A is a 3x3 nilpotent matrix with an index of nilpotence $k$ of 3 so that $A^k = 0$ and $A\vec{x}$ and $A^2\vec{x}$ is non-...
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Is this linear independence proof valid?

Let's say we have a 4x4 nilpotent matrix $V$ with an index of 4 and a vector $\vec{x}$ which is an element of $\mathcal{R}^4$, matrix $V$ and $\vec{x}$ are non-zero. How would we show that $V\vec{x}$ ...
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Application of Fitting's Lemma

Let $R$ be a ring. Let $M$ be an indecomposable $R$-module with length $n$. Let $f\in\operatorname{End}(M)\setminus\operatorname{Aut}(M)$. Show that $f^n=0$ What I have so far: We can apply Fitting's ...

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