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Questions tagged [nilpotence]

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

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On $\bigg[\begin{matrix}E&O\\O&F\end{matrix}\bigg],B_b=\left[\begin{matrix}2b&-1&0&-1\\0&b&0&0\\0&-1&0&-1\\0&1&0&b\end{matrix}\right]\in M_4(\Bbb R)$

Consider the $4\times4$ matrices $A=\bigg[\begin{matrix} E&O_2\\O_2&F \end{matrix}\bigg]$, where $E,F$ are any nilpotent $2\times2$ matrices, and $B_b=\left[\begin{matrix}2b&-1&0&-...
1
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2answers
49 views

Proof of Jordan-Chevally-Decomposition

Let A be a square matrix over $\mathbb{C}$, prove there are matrices $D$ and $N$ such that $A = D + N$ such that $D$ is diagonalizable, $N$ is Nilpotent and $DN = ND$. I can see that any ...
7
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1answer
88 views

$A$ nilpotent and $A+c_iB$ is nilpotent then $B$ is nilpotent.

Let $A$ and $B$ be $n \times n$ matrices over some field with $A$ nilpotent. Now let $c_1,\ldots,c_{n+1}$ be $n+1$ distinct scalars such that $A+c_i B$ is nilpotent for all $i=1, \ldots,n+1$. Then how ...
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3answers
71 views

The converse of “nilpotent elements are zero-divisors”

For commutative rings $A$ with identity $1\ne0$, nilpotent elements are zero-divisors. The converse is false, i.e. there is a commutative ring $A$ with identity $1\ne0$ and a zero-divisor $x$ in $A$ ...
3
votes
1answer
59 views

Nilpotent ring and Nilpotent groups.

Let $R$ be a ring (associative and with unity) and $B$ be a subring with the property that $B^n = 0$ i.e. $$ \forall\; x_1, x_2, \dots, x_n \in B: \; x_1 \cdot x_2 \cdots x_n = 0$$ My aim is to prove ...
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1answer
192 views
+200

Generalize exterior algebra: vectors are nilcube instead of nilsquare

The exterior product on a ($d$-dimensional) vector space $V$ is defined to be associative and bilinear, and to make any vector square to $0$, and is otherwise unrestricted. Formally, the exterior ...
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0answers
17 views

Nilpotent Operators Linear Algebra

Let $V$ be an $n$-dimensional vector space over an arbitrary field $K$, and let $T_1, \dots , T_n : V \rightarrow V$ be pairwise commuting nilpotent operators on V (a) Show the composition $T_1 T_2 ·...
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votes
1answer
110 views

Find a example of $A$ be $4 \times 4$ matrix such that $A$ has rank $2$ but $A^2 =0 $?

Find a example of $A$ be $4 \times 4$ matrix such that $A$ has rank $2$ but $A^2 =0 $? My attempt : $$A=\begin{bmatrix} 0 & 0 & 1 &0\\0 & 0 & 1 & 0\\0 &0 &...
4
votes
9answers
121 views

All nilpotent matrices $2 \times 2$ satisfies $A^{2}=0$ [duplicate]

I have problems to show that if $A$ is a $2 \times 2$ matrix and if there exists some positive integer such that $A^{n}=0$ then $A^{2}=0$. I only showed that $A$ is a singular matrix but nothing else. ...
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votes
2answers
88 views

Mutually commuting matrix with $A_i^2=0$

Let $A_1,\dots,A_n$ be mutually commuting $m\times m$ matrices such that $A_i^2=0$ for all $1\le i \le n$. If $m<2^n$, prove that $A_1 A_2\cdots A_n=0$ Since $A_i^2=0$ So $\operatorname{Im}(A)\...
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votes
2answers
103 views

Let $A$ and $B$ be nilpotent matrices that commute with $[A,B]$. If $A$, $B$, and $[A,B]$ are all nilpotent, show that $A+B$ is nilpotent.

How to prove that $A + B$ is nilpotent, when $A$, $B$, $[A, B]$ are nilpotent matrices, and also $A$ and $[A, B]$, $B$ and $[A, B]$ are pairs of commuting matrices? Looks like I should use binomial ...
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0answers
8 views

Irreducibility of differential operator?

Is the differential operator $D:P_n \to P_n$ is reducible? Find an element of $P_n$ that is of period $n+1$ under $D$. Here $D$ is Differential operator and $P_n$ is the vector space of all ...
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3answers
45 views

Show that every maximal ideal of $R$ contains the element $a$.

Let $R$ be a commutative ring with identity and let $a\in R$ such that $a^n=0$, for some positive integer $n$. Suppose that $I$ is and ideal of $R$. Define $(I,a)=\{x+ra\ :\ x\in I\ \text{and}\ r\in R\...
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1answer
49 views

Find Jordan Decomposition without calculating $\ker(A - \lambda E)$

Without calculating the null space of $(A - \lambda E)$, find the Jordan decomposition of $$ A = \begin{pmatrix} 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 0 &...
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0answers
73 views

Show that two nilpotent matrices are closed under addition and multiplication

From Serge Lang's linear algebra textbook: A square matrix $A$ is said to be nilpotent if $A^r=O$ for some integer $r \geq 1$. Let $A$, $B$ be nilpotent matrices, of the same size, and assume $AB$...
2
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1answer
57 views

If $G$ is a nilpotent group of class $2$, show $G' ≤ Z(G)$

If $G$ is a nilpotent group of class $2$, show $G' ≤ Z(G)$, and here we are taking the definition that class of nilpotency is the smallest length of the central series of $G$; where $G'$ is the ...
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1answer
41 views

Show that if $R$ is a ring with a single ideal prime minimal $\neq(0)$ then $R$ has non-trivial nilpotent elements

Show that if $R$ is a ring with a single ideal prime minimal $\neq(0)$ then $R$ has non-trivial nilpotent elements, that is, elements to $a\neq 0$ for those who $a^n=0$, for a certain power $n> 0$. ...
6
votes
2answers
65 views

Nilpotent elements of group algebra $\Bbb CG$

Goal: explicitly find a nilpotent element of the group algebra $\Bbb C G$ for some finite group $G$. This exists if and only if $G$ is non abelian by Maschke's theorem and Wedderburn-Artin. By ...
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1answer
36 views

“Determine the zero divisors, nilpotents and the non-invertibles in $ \mathbb{Q}[x] /(x^{4}-2x^{3}+3x^{2}-4x+2)$”

Good evening, I would like to ask the following exercise : "Determine the zero divisors, nilpotents and the non-invertibles in $ \mathbb{Q}[x] /(x^{4}-2x^{3}+3x^{2}-4x+2)$" I saw that the ...
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0answers
37 views

Nilpotent binary matrices over finite fields

I am studying on nilpotent matrices over finite fields. By definition a square matrix $A$ is $p$-nilpotent if a power of $A$ modulo $p$ is the zero matrix. For example. Let $J_2$ be the $2\times 2$ ...
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1answer
37 views

Nilpotent Jacobson radical

I have to prove the following: If $R$ is a finite-dimensional algebra over a field $F$, then $J(R)$ is nilpotent. I thought about this, but there are some gaps: Because $R$ is in particular a ...
1
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0answers
40 views

when the finite nilpotent group is cyclic?

In the notion of nilpotent group, a finite group $G$ is nilpotent iff every sylow subgroups is nomral in $G$ iff $G$ is isomorphic to direct products of all sylow subgroups of $G$ I wonder in what ...
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votes
2answers
45 views

Rank is less than $\frac n2$

Let $A \in \mathbb{R}^{n \times n}$ such that $A^2 =0$. Prove that $\mbox{rank}(A) \leq \frac n2$. With Cayley-Hamilton, the characteristic polynomial is $\chi_A=X^2$. I also know $\dim A = \dim(Im(A)...
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0answers
26 views

NxN nilpotent matrix, d-dimensional eigenspace

The set of matrices described in the title form a set in $R^{N\times N}$. What is the dimension of this set? For example, if N=2 and d=2, the only matrix is the zero matrix so the dimension is zero. ...
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0answers
65 views

Nilpotent Maps (intuition)

My intuition of a nilpotent map is that it is a map for which some Vectors are cycled through all linearly independant vectors, until it hits one which is in the kernel, and then it collapses. The ...
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0answers
62 views

Identifying nilpotent $2\times 2$ matrices with a quadratic cone

Recall that $A \in \operatorname{Mat}(n, \mathbb{R})$ is called nilpotent if there exists n > 0 such that $A^n = 0.$ Observe that if A is nilpotent then its characteristic polynomial is, up to a sign,...
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4answers
36 views

If matrix $A$ is similar to $B$ and $A$ is nilpotent, does that imply that $B$ is also nilpotent

If matrix $A$ in $\mathbb{R}^{n\times n}$ is nilpotent, what I know: all the eigenvalues of $A$ are $0$ the determinant of $A$ is $0$ (because the minimal polynomial has to be $0$) the rank of $A$ ...
1
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2answers
41 views

eigenvalues of nilpotent matrix problem

I have this following problem: let $0\neq A\in M_n(F)$ such that $A^k=0$ for integer $k>1$ a. what are the eigenvalues of A? b. show that $B=\alpha\cdot I_n-A$ invertible for every scalar $\...
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0answers
26 views

condition for positivity

I am end up with an expression of a solution of an algebraic equation $N\dot x(t)= x(t)+Bu(t)$ $x(t)=-[B\hspace{0.2cm} NB \hspace{0.2cm} N^2B\dots N^{m-1}B]_{n\times mn}\begin{bmatrix}u(t)\\u^{(1)}(t)...
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1answer
34 views

Is the nilradical of $\Bbb Z/n\Bbb Z$ equal to that of $(\Bbb Z/n\Bbb Z)/N$ where $N=\sqrt{(0)}$?

Question: Determine the nilradical $N := 􏰄\sqrt{(0)}$ in $\Bbb Z/n\Bbb Z$ and in $(\Bbb Z/n\Bbb Z) /N$ . My attempt: $􏰄\sqrt{(0)}=\{x+n\Bbb Z\in\Bbb Z/n\Bbb Z\ |\ \exists m\in\Bbb N\ x^m\in n\Bbb Z\...
2
votes
2answers
50 views

If $\not\exists r+I$ nilpotent then $R/I$ is an integral domain

Let $R$ be a commutative ring and $I$ an ideal of $R$. Definition: a non zero element $ a\in R$ is nilpotent if $\exists n\in\Bbb N$ s.t. $a^n=0$ It is true that if $R/I$ is an integral domain then ...
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1answer
32 views

Let R be a prime Jacobson ring (with 1) [closed]

How to show that R has no nonzero nil ideals? Any hint would be great.
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1answer
72 views

Prove that this linear transformation is nilpotent

The Question: Let $V$ be a finite dimensional vector space over $\Bbb C$, with linear map $T:V \rightarrow V$. Suppose that the minimal polynomial of $T$ is $$m_T(x) = p(x)q(x) \\ p(x)=(x-\lambda)^\...
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votes
2answers
29 views

Nilpotent/ Quotient

I am little fuzzy on how can we conclude that if $L$ is a Lie algebra, and if $L/Z(L)$ is nilpotent, then so is L. Where $Z(L)$ is the center of $L$, namely $Z(L) = \{z \in L \vert [xz] = 0 \ for \ ...
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votes
4answers
53 views

Eigenvalue problem

If all the eigenvalues of $2 \times 2$ matrix $A$ are zero then prove that $A^2=0$. $0$ is an eigenvalue of $A$. Then $AX=0$, again $A^2X=0$. Then $X$ is an eigenvector of both $A$ and $A^2$. Can we ...
3
votes
0answers
43 views

For all $A, B \in \mathcal{M}_n(\mathbb{C})$, if $[A, [A, B]] = 0$, then $[A, B]$ is nilpotent [duplicate]

$\newcommand{\tr}{\mathrm{Tr }}$ Let be $\mathcal{M}_n(\mathbb{C})$ the set of complex-valued square matrices of order $n$ and $[A, B] = AB - BA$ a commutator. If, for some $A, B \in \mathcal{M}_n(\...
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2answers
54 views

Can the operator $\operatorname{id}_V-(AB-BA)$ be nilpotent in the infinite-dimensional case.

Let $V$ be a infinite-dimensional vector space over the field of characteristic $0$ and $A,B$ be linear operators of $V$. Let $\operatorname{id}_V$ be an identical operator. Using trace function it ...
3
votes
1answer
27 views

g+h is an isomorphism (nilpotent linear map)

$V$ is a vector space, $g: V \to V$ an isomorphism and $h: V \to V$ a nilpotent linear map. Also, $g$ and $h$ commute. This implies that $g+h$ is an isomorphism. Well, $h$ is nilpotent so $h^n=0$. ...
1
vote
0answers
59 views

On the central series of the group of unitriangular matrices over a commutative, unitary ring

Let $R$ be a commutative ring with identity and put $U=U(n,R)$, the group of $n\times n$ (upper) unitriangular matrices over R. Define $U_i$ to be the be the subgroup of $U$ having (at least) $i-1$ ...
2
votes
2answers
103 views

Sums of Nilpotent Matrices

Let $A =$ diag$(a_1,a_2,…,a_n)$, where the sum of all $a_i$’s is zero. Show that A is a sum of nilpotent matrices. My idea: $\begin{bmatrix} 2 & 0 \\ 0 & -2 \end{bmatrix}$ = $\begin{bmatrix}...
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0answers
40 views

Proving a ring $A$, generated by Noetherian subring and nilpotent element, is Noetherian again.

I am studying some algebra during my spare time. In particular I am learning about Noetherian rings. A friend sent me the following excersise, and I am not able to solve it. Suppose that a ring $A$ ...
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votes
4answers
124 views

Finding Jordan Basis of a nilpotent matrix

I have to find the basis of the nilpotent matrix $A$ of size $3\times3$. $$ \left(\begin{matrix} 1&1&1\\-1&-1&-1\\1&1&0\end{matrix}\right) $$ I found that $A^3=0$ $ker A^0=...
6
votes
2answers
161 views

For matrix $A$, if $A^4 = 0$ does this also mean that $A^2 = 0$?

For an arbitrary matrix $A$, if $A^4 = 0$ does this also mean that $A^2 = 0$? My thinking is that it does since I can reduce $A^4$ into $(A^2)^2$ but I'm not sure if this helps or not.
1
vote
1answer
31 views

Idempotents over a ring with zero divisors

Let $R = \mathbb{Z}/4\mathbb{Z} = \{0, 1, 2, 3\}$ and the group $G = \mathbb{Z}/2\mathbb{Z} = \{e, a\}$. Consider the group ring $R[G]$. I have read somewhere (1) that $\frac{e + a}{2}$ and $\frac{e -...
2
votes
0answers
48 views

Prove that the operator is nilpotent

Let $V$ be a finite-dimensional vector space and $T:V\rightarrow V$ be an operator such that $tr(\Lambda^q(T))=0$ for all $q$. Prove that $T$ is nilpotent. I would like to understand the following ...
1
vote
4answers
40 views

Showing a Matrix is Nilpotent of a certain degree using only properties of matrix multiplication/summations

Suppose I had an $n \times n$ matrix $$N = \begin{pmatrix} 0 & 1 & 0 & 0 & \cdots & 0 & 0\\0 & 0 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & 1 &...
1
vote
1answer
48 views

Can any linear map, $\phi$, can be expressed a as $\phi = S + N - N^T,$ where $S$ is a self-adjoint and $N$ is a nilpotent map

Inspired from this and this , I claim that Any linear map, $\phi$ on a real vector space $E$, can be written as a $$\phi = S + N - N^T,$$ where $S$ is a self-adjoint and $N$ is a nilpotent ...
1
vote
1answer
52 views

In a commutative algebra, every nil ideal is nilpotent, right?

Can anyone prove or disprove the following: In a commutative algebra, every nil ideal is nilpotent. The reason I am asking is that I have an ideal which I can show to be nil ideal but which might not ...
1
vote
1answer
111 views

Nilpotent matrix and direct sum

Let $A$ be a nilpotent matrix and suppose $A^6 = 0$ but $A^5 \neq 0$. Further, suppose the vector spaces $\mathcal V_1, \ldots, \mathcal V_6$ satisfy $$\ker(A^{i}) = \mathcal V_1 \oplus \mathcal V_2 \...
0
votes
2answers
60 views

Rank of a matrix from its order of nilpotency

Let $A$ be an $n \times n$ nilpotent matrix of order $r$, i.e. $A^r = 0$ ,but $A^{r-1} \neq 0$. Can we conclude from the above information about the rank of $A$? Can we say rank $A$ is at most $r$?