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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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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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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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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