# 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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### 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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### 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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### 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 ...