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Questions tagged [nilpotence]

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

3
votes
2answers
94 views

Decomposing a symmetric matrix as a sum of nilpotent matrices

Assume that a real-valued symmetric matrix $M$ with trace zero can be written as $$ M = A + A^T, $$ with $A^2=0$. Given that $M$ is known, how (if possible) can $A$ be found? The diagonal elements ...
1
vote
1answer
34 views

Help with proof of $ \mathbb{C}[X] \simeq R $ where $R$ is a $ \mathbb{C}$-algebra without nilpotents

I am trying to understand the proof of the following proposition: Let $X \subset \mathbb{A}^n$ be closed. Let $ R $ be a finitely generated $ \mathbb{C}$-algebra without nilpotents. There exists an ...
2
votes
1answer
38 views

How to find a Jordan basis and a Jordan matrix for a nilpotent matrix?

I am trying to find a general step-by-step "easy" / "intuitive" solution to finding Jordan basis and Jordan matrix (based on the basis) for a nilpotent matrix. If you can add an intuition for the ...
0
votes
3answers
33 views

how does a nilpotent triangular matrix operate

I can only find specific examples online,and I would like to know in general: Given an $n \times n$-matrix \begin{align} A = \begin{pmatrix} 0 & b & 0 &\cdots & 0 \\ ...
0
votes
1answer
22 views

Quotient algebras of nilpotent Lie algebra are nilpotent

For the following proposition I found a proof in some notes that I don't understand. Below definition 1 defines the terminology I'm using, and proof attempt 1 gives my attempt at the proposition. ...
3
votes
1answer
111 views

Show that $AB$ is nilpotent

Let $A,B$ be two $n\times n$ matrices. If $A^2B-2ABA+BA^2=0$ and $A$ is nilpotent, that is, there exists a positive integer $k$ such that $A^k=0$. Show that $AB$ is nilpotent. If $k=2$, it is OK. ...
7
votes
0answers
197 views

If nilpotent matrix $A$ and $AB−BA$ commute, show that $AB$ is nilpotent.

Let $A$ and $B$ be $n×n$ complex matrices. If $A$ is a nilpotent matrix, and $A$ commute with $AB−BA$, show that $AB$ is nilpotent. Equivalently, the question can be expressed as following ...
2
votes
1answer
67 views

When is the matrix $I-P$ nilpotent?

Note: sorry for the long post. My question is in the second quote. A few years ago, I saw the following problem on Facebook. I would like to ask a generalised version of this problem. Here is the ...
0
votes
1answer
22 views

If $H$ is a finite $p$-group and $J_H = \ker(k[H] \rightarrow k)$, then $J_H$ is nilpotent

Suppose $H$ is a finite $p$-group, $k$ a field of characteristic $p$, and consider the map $k[H] \rightarrow k$ via $h \rightarrow 1, \; \forall h \in H$, and extending $k$-linearly. Let $J_H$ be the ...
0
votes
1answer
21 views

nilpotent endomorphism and $Im(f)+Ker(f) \neq dim(V)$

If I have an endomorphism between vector spaces $f:V \rightarrow V$, such that $Im(f)+Ker(f) \neq dim(V)$, is this equivalent to $f$ being nilpotent?
8
votes
1answer
183 views

If $B$ is nilpotent and $AB=BA$ then $\det(A+B) = \det(A)$

The following stumps me: Let $\mathbb K$ be a field. Let $A, B \in \mathbb K^{n \times n}$ where $B$ is nilpotent and commutes with $A$, i.e., $A B = B A$. Show that $$ \det(A+B)=\det(A) $$ I have ...
2
votes
2answers
94 views

Determine all $2 \times 2$ real matrices $A$ such that $(1) \ \ A^2=I$, $(2) \ \ A^2=0$

Determine all $2 \times 2$ real matrices $A$ such that $(1) \ \ A^2=I$, $(2) \ \ A^2=0$ I came across this problem recently where I have to determine all the $2\times2$ matrices satisfying the ...
1
vote
1answer
40 views

Nilradical is the intersection of finitely many minimal prime ideals.

Let $R \neq \{0 \}$ be a commutative ring with identity. Suppose that $R$ has only finitely many minimal prime ideals $p_1,\dots , p_s.$ Then $$\sqrt {0} = \bigcap\limits_{i=1}^{s} p_i.$$ I ...
2
votes
1answer
29 views

Exericse about linear map $T\in L(V)$, where $\dim V=n\geq2$, with $\operatorname{null}T^{n-1}\neq\operatorname{null}T^n$

I have this problem that I am attempting, and am struggling with (b). -- Assume $\dim V = n \geq 2$ and that $T \in L(V)$ such that $\operatorname{null}T^{n-1}\neq\operatorname{null}T{^n}$ -- (a) ...
1
vote
1answer
66 views

Nilpotent, Idempotent and Involutory Matrix

With exception of the zero matrix, can a matrix be nilpotent $(A^k=0)$ and idempotent $(A^2=A)$ at the same time? and With exception of the identity matrix, can a matrix be idempotent and involutory ...
1
vote
1answer
41 views

Usage of Zorn's Lemma to prove that the intersection of all prime ideals contains only nilpotent elements.

I have read a couple proofs that that the intersection of all prime ideals contains only nilpotent elements that use a claim like this: Suppose that $a$ is an element of $A$ that is not nilpotent. ...
0
votes
1answer
91 views

Proving that an element is nilpotent [closed]

Let $m,n \in \mathbb N, n \geq 2$. $A$ is a ring with $|A|=n$ and $a \in A$ s.t. $1-a^k$ is invertible $\forall k \in \{m+1,m+2,...,m+n-1\}$. Prove that $a$ is nilpotent. Can somebody help me, ...
0
votes
1answer
45 views

k-th power nilpotence ($k\ge 3$)

Inspired by my solution to 2x2 Matrix with no zero entries where $A^k=0$ - Nilpotence? I came up with this problem. Let $A$ be a $nxn$ matrix which is not the zero matrix $0$ (in which all elements ...
0
votes
4answers
56 views

2x2 Matrix with no zero entries where $A^k=0$ - Nilpotence?

Find an example of a $2x2$ matrix $A$ that has no zero entries but is such that $A^K=0$ for some positive integer k. Here is my thinking: When $k=1, A=0$, but this contradicts that the matrix has no ...
0
votes
1answer
8 views

Let B be a nilpotent linear operator with index p. Is operator $ N^2 + \alpha*N $ nilpotent?

Let B be a nilpotent linear operator with index p. Is operator $ N^2 + \alpha*N $ nilpotent? ( $ \alpha \in \mathbb{R}$) I think that when $ \alpha = 0$ then the index is $p/2$ for even p, but when p ...
6
votes
3answers
579 views

Eigenvalues of a matrix whose square is zero

Let $A$ be a nonzero $3 \times 3$ matrix such that $A^2=0$. Then what is the number of non-zero eigenvalues of the matrix? I am unable to figure out the eigenvalues of the above matrix. P.S.: how ...
0
votes
1answer
35 views

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
vote
2answers
91 views

Proof of Jordan-Chevalley 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 nilpotent ...
6
votes
1answer
109 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 ...
2
votes
3answers
100 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 $A$ ...
4
votes
1answer
91 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 ...
18
votes
1answer
290 views

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 ...
1
vote
0answers
35 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 $...
0
votes
1answer
200 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
163 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. ...
6
votes
2answers
105 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)\...
2
votes
2answers
168 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 ...
0
votes
0answers
15 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 ...
2
votes
3answers
63 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\...
1
vote
1answer
57 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 &...
0
votes
0answers
141 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
votes
1answer
73 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 ...
1
vote
1answer
46 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
78 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 ...
0
votes
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 ...
1
vote
0answers
45 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 $2\times 2$ ...
1
vote
1answer
93 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
vote
0answers
42 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 ...
0
votes
2answers
47 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)...
0
votes
0answers
30 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. ...
4
votes
0answers
77 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 ...
1
vote
0answers
68 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,...
0
votes
4answers
39 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
vote
2answers
55 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 $\...
0
votes
0answers
32 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)...