A nilpotent element of a ring has $a^n=0$ for some integer $n$.
A nilpotent element of a ring has $a^n=0$ for some integer $n$.
For example, in the ring $\mathbb{M_{2\times2}}$, $\begin{pmatrix}0&1\\0&0\end{pmatrix}$ is nilpotent with degree $2$.
Questions tagged [nilpotence]
640 questions
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votes
2 answers
What do zero eigenvalues mean?
What is the geometric meaning of a $3 \times 3$ matrix having all three eigenvalues as zero? I have interpretations in mind for $0$, $1$, and $2$ eigenvalues being zero, but what about all of them?
ThanksABundle
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30
votes
2 answers
Direct proof that nilpotent matrix has zero trace
Does anyone know a proof from the first principles that a nilpotent matrix has zero trace. No eigenvalues, no characteristic polynomials, just definition and basic facts about bases and matrices.
Norbert
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27
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2 answers
Prove that the only eigenvalue of a nilpotent operator is 0?
I need to prove that:
if a linear operator $\phi : V \rightarrow V$ on a vector space is nilpotent, then its only eigenvalue is $0$.
I know how to prove that this for a nilpotent matrix, but I'm not sure in the case of an operator. How would I be…
Mathlete
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votes
4 answers
Why are eigenvalues of nilpotent matrices equal to zero?
If $A$ is a $ \displaystyle 10 \times 10 $ matrix such that $A^{3} = 0$ but $A^{2} \neq 0$ (so A is nilpotent) then I know that $A$ is not invertible, but why does at least one eigenvalue of $A$ have to be equal to zero? How would one show that…
David
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20
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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 algebra $\Lambda V$ is a quotient of the tensor…
mr_e_man
- 5,986
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4 answers
Prove that $A+I$ is invertible if $A$ is nilpotent
Possible Duplicate:
Units and Nilpotents
Given $A^{2012}=0$ prove that $A+I$ is invertible and find an expression for $(A+I)^{-1}$ in terms of $A$. ($I$ is the identity matrix).
ugoolm
- 199
18
votes
4 answers
Upper bound for the rank of a nilpotent matrix , if $A^2 \ne 0$
I came across the fact that the rank of a $n \times n$-matrix $A$ with $A^2=0$ is at most $\frac{n}{2}$. The easiest way to proof this is using the inequality $\operatorname{rank}(A) + \operatorname{rank}(B) - n \leqslant \operatorname{rank}(AB)$.…
Peter
- 86,576
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4 answers
$A$ is normal and nilpotent, show $A=0$
Given a matrix $A \in R^{n \times n}$ which is normal ($AA^H=A^HA$ where $A^H$ is hermitian of $A$) and nilpotent ($A^k=0$ for some $k$). Now we need to show that $A=0$.
(This is essentially exercise 5(b) in sec. 80 on p.162 of Paul R. Halmos'…
Learner
- 2,796
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2 answers
Lifting idempotents modulo a nilpotent ideal
The problem is this:
Suppose $I \subseteq R$ is a nilpotent ideal and there is $r \in R$ with $r \equiv r^2 \pmod I$. Show $r \equiv e \pmod I$ for some $e \in R$ idempotent.
I have spent a few hours rolling around in abstracta with no…
user309475
15
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4 answers
How to show that the nth power of a $n \times n$ nilpotent matrix equals to zero $A^n=0$
$A$ is a $n\times n$ matrix such that $ A^m = 0 $ for some positive integer $m$.
Show that $A^n = 0$.
My attempt:
For $n > m$, it's obvious since matrix multiplication is associative.
For $n < m$, $A^n\times A^{m-n} = 0$; not sure what to do…
Geralt of Rivia
- 1,274
12
votes
2 answers
Nilpotent matrix and relation between its powers and dimension of kernels
Given a 4x4 matrix $T$ over $\mathbb{R}$ such that $T^4 = 0 $, $k_i = \textsf{dim} Ker(T^i)$, I need to check which of the following sequences, $$k_1\leq k_2 \leq k_3 \leq k_4,$$ is NOT possible :
$ 1)\; 1\leq 3 \leq 4 \leq 4$
$2) \;…
Vishesh
- 2,978
11
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1 answer
Looking for a counterexample when dropping one of the constraints (Linear algebra)
I want to find a counterexample for the following "Theorem":
Let $V \neq 0$ be a finite dimensional $K$ - vector space and $L \subset \mathfrak{gl}(V)$ a linear subspace. If all $x \in L$ are nilpotent as maps $V \rightarrow V$ then there is a $v…
tohann123
- 317
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0 answers
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 description.
Let $A$ and $B$ be $n×n$ complex…
Gardenia625
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- 1
- 7
10
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4 answers
If A+tB is nilpotent for n+1 distinct values of t, then A and B are nilpotent.
Suppose $A$ and $B$ are $n\times n$ matrices over $\mathbb{R}$ such that for $n+1$ distinct $t \in \mathbb{R}$, the matrix $A+tB$ is nilpotent. Prove that $A$ and $B$ are nilpotent.
What I've tried so far:
Define $f(t)=(A+tB)^n$. Then $f(t)$…
Q-rious
- 175
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10
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The number of $3\times 3$ nilpotent matrices over $\mathbb{F}_q$ using the orbit-stabilizer theorem
The Fine-Herstein theorem says that the number of nilpotent $n\times n$ matrices over $\mathbb{F}_q$ is $q^{n^2-n}$. I am trying to verify this for the case $n=3$ using the orbit-stabilizer theorem. Here is my method:
Knowing that the minimal…
Justine
- 1,103