Questions tagged [idempotents]

For questions about elements which satisfy $x\cdot x=x$ where $\cdot$ is a composition law.

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$S$ 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 $ST=TS=0$?
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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^...
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1answer
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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 $a,b \in S$, $...
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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 $1\leq i \leq n$, ...
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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 ...
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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^...
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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 ...
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2answers
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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 ...
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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 ...
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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 ...
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1answer
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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 \...
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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 ...
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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 ...
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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,......
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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 ...
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1answer
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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= ...
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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 \...
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1answer
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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=\{...
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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 ...
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1answer
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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 ...
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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 ...
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1answer
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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 \...
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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 \...
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25 views

Matrices Question invlving transpose

Could anyone solve this question using properties? I did it by taking a $2\times1$ matrix.
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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. ...
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2answers
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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 ...
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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(...
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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 ...
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3answers
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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 ...
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1answer
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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 $e_i \neq 0$...
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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 ...
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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....
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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,...
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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 ...
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1answer
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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 ...
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1answer
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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 ...
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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 ...
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3answers
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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 ...
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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 ...
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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 (...
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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 ...
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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 ...
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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) =...
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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 ...
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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 ...
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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 (...
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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 ...
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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 ...
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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 ...
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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 ...

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