Questions tagged [idempotents]
For questions about elements which satisfy $x\cdot x=x$ where $\cdot$ is a composition law.
366
questions
2
votes
1answer
35 views
$S$ and $T$ are idempotent linear operators in $V$ which is a vector space in $\mathbb{C}$. Prove that if $S+T$ is idempotent then $ST=TS=0$.
If $S+T$ is idempotent, then
$$(S+T)^2=S+T$$
$$\implies S^2+ST+TS+T^2=S+T$$
$$\implies S+T+ST+TS=S+T$$
$$\implies ST+TS=0$$
Now, from here how do I show that $ST=TS=0$?
1
vote
1answer
66 views
If e is idempotent, prove that 1-e is as well.
Let $(R,+,·)$ be a ring with unity. If $e$ is idempotent, prove that $1-e$ is also idempotent.
Here's my attempt:
If $e$ is idempotent then $e^n=e$. Then
$$e=e^n$$
$$e-e^n=0$$
$$e(1-e^{n-1})=0$$
$$1-e^...
0
votes
1answer
21 views
In a finite semigroup, does each ideal (esp. the minimal ideal) have an idempotent?
Given a finite semigroup $S$, let $I$ be a $\mathcal{J}$-minimal ideal, where $\mathcal{J}$ is equivalence relation brought about by the Green relation $\leq_{\mathcal{J}}$, where given $a,b \in S$, $...
3
votes
2answers
37 views
Orthogonal idempotents in $M_{4\times 4}(\mathbb{C})$.
If $e_1,\cdots,e_n \in M_{4\times 4}(\mathbb{C})$ are $n$ distinct, nonzero, $4\times 4$ matrices with complex entries that satisfy $e_ie_j = e_je_i = 0$ and $e_i^2 = e_i$ for all $1\leq i \leq n$, ...
1
vote
0answers
33 views
Idempotence of $X(X'X)^{-1}X' - \frac 1 n J$
I am facing the following problem in the context of linear regression:
On the one hand, it holds that $SS_{reg}=\sum(\hat{Yi} - \bar{Y})^2= Y'(H - \frac 1 nJ)Y$ where $H = X(X'X)^{-1}X'$ and $J$ is a ...
0
votes
1answer
29 views
How to construct a symmetric and idempotent matrix from given two non-colinear vectors?
Given a vector $x\in \Bbb R^n$ it is easy to see that $A:=\dfrac{xx^t}{x^tx}$ is symmetric and idempotent. i.e. $A^t=\dfrac{(xx^t)^t}{x^tx}=\dfrac{((x^t)^t(x)^t)}{x^tx}=A$ and $A^2=\dfrac{xx^txx^t}{x^...
1
vote
0answers
16 views
Idempotent Matrix is a Projection on Two Subspaces
Let V be a finite dimensional vector space and $T \in L(V)$. Show that
a) if T^2 = T, then T is diagonalisable with eigenvalues 0 and 1, V = V1⊕V2 for some subspaces V1, V2 such that T is the ...
2
votes
2answers
84 views
Proof of Idempotency for Matrices
I'm currently looking at achieving some proof that an $n \times n$ matrix (let's say $Y$) is idempotent if and only if $I_n-Y$ is an idempotent matrix, (with $I_n$ being the identity matrix).
I have ...
0
votes
1answer
83 views
Idempotency in Matrices and deducing Linear Map Compositions
Hello, could anyone help me out with these two questions.
With question d) I know that because D is an idempotent matrix, then In - D must also be idempotent through:
(In-D)(In-D) = In-2Dn+D = In - D
...
0
votes
1answer
107 views
Matrix Idempotency Questions
Firstly I apologise for the use of an image but I am still unfamiliar with using the math text on here.
So I am currently studying linear algebra at second year level and have been working on this ...
2
votes
1answer
44 views
Idempotent Endomorphism of semi simple module
I try to show the following exercise from Group Representation Theory Book from Peter Webb:
Let $A$ be a ring with $1$. Let $V$ be a semisimple $A$-module with finitely many simple summands and $e,f \...
2
votes
3answers
68 views
Is the set of all Idempotent matrix in $M_n(\mathbb{F})$ linearly independent?
Is the set $I:=\{\text{set of all Idempotent matrix in}\ M_n(\mathbb{F})\}$ linearly independent?
My thought: I think the answer is no if $\mathbb{F}$ is infinite.
If $\mathbb{F}$ is infinite then the ...
2
votes
0answers
36 views
Sufficient Criterion for Word Equivalence in Idempotent Semigroups
I am reading Baader's paper "The Theory of Idempotent Semigroups is of Unification Type Zero" and I have a doubt about a property of idempotent semigroups.
Let $AI = \{(xy)z = x(yz), x^2 = x ...
0
votes
0answers
77 views
Prove that the set of idempotent matrices generates $M_n(F)$.
Attempt: Firstly, we note that the $n \times n$ matrices of the form $c_{ij}=1$ for $ (i,j)=(r,r)$, and zero everywhere else are idempotent matrices. [For eg, the diagonal matrix $\text{diag}(0,1,0,......
0
votes
0answers
34 views
Is there some classification of idempotent pseudo-differential operators?
Let $\pi:E\rightarrow M$ be a complex vector bundle over a smooth compact manifold $M$. Let $\operatorname{Psd}(E)$ be the associative $\mathbb{Z}$-graded complex algebra of pseudo-differential ...
0
votes
1answer
49 views
How to get $i = ij + i^2$
$A$ is a $k$-algebra where $k$ is a commutative ring.
Then for $e$ an idempotent in $A$ and $U,V$ submodules of $Ae$ s.t. $Ae = U\oplus V$, there is unique idempotents $i, j$ in $A$ s.t. $U = Ai, V= ...
0
votes
0answers
16 views
Least squares with an idempotent matrix
This question is mainly to confirm my intuition (since my proof is counter intuititive).
Suppose I have a least squares formulation, $AX=B$ where $A \in \mathbb{R}^{d \times d}$, $X \in \mathbb{R}^{d \...
5
votes
1answer
120 views
Does every involution have a fixed point?
I was trying to find functions $f:X\to X$ such that $f^2$ is the identity on $X$ and such that $f(x)\ne x$ for all $x\in X$. First, this is a bit vague, of course you can take any two-element set $X=\{...
1
vote
2answers
169 views
Idempotent linear operators are projections
I'm working on this problem:
Suppose $V$ is a Hilbert space and $P : V \to V$ is a linear map such that $P^2= P$ and $\Vert Pf \Vert \leq \Vert f \Vert$ for every $f \in V$. Prove that there exists a ...
2
votes
1answer
33 views
Equivalency of the maximum and minimum conditions of idempotents of a ring
Let $R$ be a ring with unit, and let $I$ be the set of all idempotents of $R$, that is, all $e\in R$ such that $e^2 = e$. We put a partial ordering $\leq$ on $I$ by saying $e\leq f$ if $ef=e=fe$ or ...
0
votes
1answer
22 views
Given a idempotent endomorphism, show that $M = \ker(f) \oplus \ker (f-\text{id}_M)$
Let $M$ be an A-module and $f \in \text{End}(M)$ an idempotent endomorphism. Show that
$M = \ker(f) \oplus \ker (f-\text{id}_M)$.
I don't know how to approach this exercise, using only properties of ...
1
vote
1answer
19 views
Find idempotent given generator poly and check poynomial by Bezout algorithm
I have the cyclic code $C$ of length $8$ and dimension $4$ over $\mathbb{F}_3$ and with check polynomial $$g(x) = (x-\alpha)(x-\alpha^2)(x-\alpha^3)(x-\alpha^6) = x^4+x^3+x+2$$ where $\alpha \in \...
0
votes
0answers
36 views
On Hilbert space, for projections $P,Q$, $P+Q$ is a projection if and only if $\text{ran}P\perp \text{ran}Q$
I am going through Functional analysis text by J.Conway, and encountered with next problem (2.3.4) :
Let $P$ and $Q$ be projections. Show $P+Q$ is a projection if and only if $\text{ran}P\perp \...
0
votes
0answers
25 views
Matrices Question invlving transpose
Could anyone solve this question using properties?
I did it by taking a $2\times1$ matrix.
2
votes
3answers
145 views
All idempotent elements in $M_n(F)$
I know that every idempotent matrix is diagonalizable. Thus, I can search diagonal matrices for all the idempotent elements. In this case, $(n,0), (n,1),..,(n,n)=2^n$ diagonal matrix can be obtained. ...
1
vote
2answers
41 views
Prove that if A is idempotent and fullfills $A = A^{-1}$ then it follows that $A = I_n$
I'm currently learning linear algebra and I have stumbled across the following example in my book without a solution.
A matrix $B \in \mathbb {R}^{n x n} $ is called idempotent if $BB = B$. Prove ...
0
votes
0answers
37 views
Primitive idempotents in symmetric group rings
I am reading on a paper about the character theory of symmetric groups $S_n$. There is a claim:
Set $\sigma$ be an idempotent in the ring $R(S_n)$. Then $\sigma$ is primitive iff for any $r\in R(...
1
vote
1answer
32 views
Representation of non-primitive idempotent in a finite dimensional algebra
I want to prove the following statement:
For any non-primitive idempotent in a finite dimensional algebra, it can be represented as a sum of orthogonal primitive idempotents.
I tried to decompose ...
2
votes
3answers
96 views
Does $P \circ P =P$ and $\langle Px, y \rangle = \langle x, Py \rangle$ imply $P$ is linear?
Let $(H, \langle \cdot, \cdot \rangle)$ be a Hilbert space and $P:H \to H$. Suppose that
$P \circ P =P$
$\langle Px, y \rangle = \langle x, Py \rangle$ for all $(x,y) \in H^2.$
I ...
-1
votes
1answer
43 views
How to show $f$ is surjective?
Let $R$ be a ring. An element $e \in R$ is called idempotent iff $e^2=e$.
If $R$ is an integral domain, then $0,1$ are the only idempotent elements.
If $1=e_1+\cdots e_s$ with elements $e_i \neq 0$...
1
vote
1answer
85 views
Idempotent matrix of the form $(D-A)$
Does there exist an idempotent matrix of the form $P=(D-A)$ where $P^2 = P$ if $A$ is idempotent? $D$ is a diagonal matrix with positive distinct entries. For the trivial case when $D$ is the identity ...
1
vote
1answer
83 views
Hilbert space self adjoint and idempotent
A linear operator $P: H \to H$ on a Hilbert space $H$ is self-adjoint if for every $f, g \in H$, $\langle Pf, g\rangle=\langle f,Pg\rangle$ and is idempotent if for every $f \in H$, $P(P(f))=P(f)$ (i....
3
votes
2answers
150 views
If a matrix is positive-semidefinite, is Hermitian, and has a trace of 1, is it idempotent?
I have a matrix $A^{n\times n}$ that is positive semi-def. and Hermitian. Also, $Tr(A) = 1$.
I want to show that $A$ is idempotent. Are these properties enough? If so, how would one show this? If not,...
1
vote
1answer
32 views
If $P,Q$ orthogonal projections on $H$ and $T\colon H\to H$ extension of Hilbert isomorphism $S\colon P(H)\to Q(H)$, then $Q=TT^{*}$.
Let $P,Q\colon H\to H$ orthogonal projections (i.e. $P^{2}=P^{*}=P$ and $Q^{2}=Q^{*}=Q$) on a Hilbert space $H$ and let $S\colon P(H)\to Q(H)$ be an isomorphism of Hilbert spaces. We can extend $S$ to ...
3
votes
1answer
28 views
If $p^{2}=p^{*}=p$ and $pa^{*}a=paa^{*}=0$, then $a^{*}p+pa=0$.
Suppose that $p$ is a projection (i.e. $p^{2}=p^{*}=p$) in a C*-algebra $A$. Let $a\in A$ be an element such that $pa^{*}a=paa^{*}=0$. I want to prove that
$$a^{*}p+pa=0.$$
I tried to express $a$ in ...
1
vote
1answer
15 views
Proof in idealization of almost clean ring.
Found this on Weakly Clean Rings and Almost Clean Rings by Ahn & Anderson.
Definition 2.1. A ring $R$ is almost clean if each $x\in R$ can be written as $x=r+e$ where $r\in reg(R)$, the set of ...
1
vote
1answer
49 views
Find idempotents in a set of mappings
Find idempotent elements of $S$ the set of maps $f:X \rightarrow R$ where $X$ is a given set and $R$ is a given ring with the operations defined by $(f+g)(r)=f(r)+g(r)$ and $(fg)(r)=f(r)g(r)$ where ...
2
votes
3answers
121 views
Idempotents in $\Bbb Z_2[x]/(x^7+1)$
I am trying to find idempotent elements in $R:=\Bbb Z_2[x]/(x^7+1)$. Of course $0,1$ are idempotents.
My attempt: For $f \in
\Bbb Z_2[x]$, let $\bar{f}$ denote its residue class. We may assume that ...
4
votes
3answers
464 views
If $A^2 = A$ then $A$ is diagonalizable
I've stumbled upon this question in my assignment:
Prove if $A_{nxn}(\mathbb C)$ with $A^2 = A$, then $A$ is diagonalizable
My first thought is to solve for $p(A)$ where $p(x) = x^2 - x$ and you ...
2
votes
1answer
38 views
Left ideals contain a non-zero idempotent
Let $D$ be a division ring and $S$ a subring of $D$ containing $1$.
Define
$$R(D,S)=\{ (x_1, x_2, \dots,x_n, s, s, \dots)\vert n \geq 1, x_i \in D, s \in S\} $$
and say that every nonzero left (...
3
votes
1answer
29 views
Adjoint action on idempotents in Clifford algebras
Given a Clifford algebra ${\operatorname{Cl}_{p,q}}$, we can define the group of units of the algebra ${\operatorname{Cl}_{p,q}}^{*}$, and the adjoint action of this group on the Clifford algebra as ...
2
votes
1answer
26 views
Do “zero-divisor-closures” of a commutative ring always contain idempotents?
Let $R$ be a commutative ring which is not necessarily a domain, and let $P(X)$ denote the power set of a set $X$.
Galois connections
Recall that for any $\rho \subseteq M\times N$, folloowing the ...
8
votes
3answers
181 views
Models of a certain (weird) equational theory
Consider the following (single-sorted) equational/algebraic theory with one binary operation symbol $\ast$ whose axioms are as follows:
$$(x \ast x) \ast (x \ast x) = x$$
$$(x \ast y) \ast (x \ast y) =...
1
vote
1answer
30 views
Zero Ideal is Indecomposable as a Finite Intersection of Primary Ideals on an Infinite Ring of Idempotents
Let R be a ring of idempotents, that is, a ring in which holds $x^2=x, \forall x \in R$. Prove that if R is infinite, then (0) is indecomposable as a finite intersection of primary ideals.
Any hints ...
2
votes
1answer
43 views
Idempotents and Direct Sums with Modules
I'm attempting to prove the following statement:
"Let $R$ be a ring, let $M$ be an $R$-module, and let $\phi:M \to M$ be an $R$-module homomorphism. Prove that if $\phi$ is an idempotent in the ...
1
vote
2answers
150 views
Karoubi envelope / idempotent completion of $R-Mod$
I have a question about motivation for building a Karoubi envelope or idempotent completion of a category $C$. A problem in a non-complete category is that it probably contains idempotent elements (...
0
votes
1answer
32 views
Monoid that is idempotent induces partial ordering
Given a commutative monoid $(M,0,\oplus)$.
Then we can define an ordering on $M$ by
$a\geq b :\Leftrightarrow \exists c: a=b\oplus c$.
The relation is then transitive and reflexive. The claim is now ...
0
votes
0answers
191 views
When is the composition of two idempotent functions itself idempotent?
If two idempotent functions f and g are individually idempotent, under what conditions is g ∘ f idempotent?
I know from this question that it is sufficient for f and g to be commutative:
If the ...
3
votes
4answers
102 views
Idempotent, commutative and invertible
Is there a mathematical class and structure in which there exist many objects that are distinct, invertible, commutative and idempotent? Like a set of toggle switches with no hysteresis, so the state ...
2
votes
3answers
177 views
Curious Case of Idempotent Matrices - Seeking a Generalisation
Here's what I initially started with:
Find a 2x2 non zero matrix $A$, satisfying $A^2=A$, and $A\neq I$.
I understand that this is fairly easy, but please keep reading for something ...
