Questions tagged [idempotents]

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

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If $M=\bigoplus_{i\in I}M_i$ and $f:M\to M$ is idempotent then $f$ induces an isomorphism between $M_i$ and $f(M_i)$.

Let $R$ be a ring, $M=\bigoplus_{i\in I} M_i$ an $R$-module with $\text{End}(M_i)$ local for all $i\in I$. Then I want to show that if $f:M\to M$ is idempotent (i.e. $f^2=f$) and nonzero, then there ...
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Show whether the scalar projection is idempotent

A while ago I asked Show whether the vector projection is idempotent, which made it clear to me that the idempotency of the vector projection could be checked with a simple substitution. Now I am ...
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Finding idempotent elements of a $C^{*}$-algebra

Let $T \in B(\mathcal{H})$ (bounded operators on a complex Hilbert space $\mathcal{H}$) and suppose $T$ is normal. Suppose furthermore $\sigma(T)=\{-1\}\cup[2,3]$. Then I want to find a complete list ...
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Find the determinant of a sum of idempotent matrices

Let $A, B \in \mathcal{M}_n (\mathbb{R})$ such that $A^2 = A$ and $B^2 = B$. If $\det(2A + B)=0$, prove that $$\det(A + 2B) = 0$$ My attempt: I know that both matrices are diagonalizable and their ...
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Idempotent relations in a ring

Let $(A,+,.)$ be a ring such that, if $x \in A$ with $6x = 0$, then $x=0$. Let $a,b,c \in A$ such that $a-b$ , $b-c$ , $c-a$ are idempotent. Prove that $a=b=c$. Unfortunately, I haven't made any big ...
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Invertible indempotent matrix not equal to identity

On Wikipedia, I found that the only non-singular idempotent matrix is $\mathrm{\mathbf{I}}$. However, the following matrix $$\begin{pmatrix} 1 & 1 & 0\\ 0 & 1 & 0 \\ 0 & 1 & 1\...
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Idempotent Semigroup $S$ with Equivalence Relation $(a R b) \iff (aba=a), (bab=b)$: $S/R$ is commutative - why?

This is exercise 11.19 (h) in Seth Warner's "Modern Algebra". We are given that $S$ is an idempotent semigroup, that is: $\forall a \in S: aa = a$. Let $R$ be the equivalence relation ...
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Let $𝐴 $ be a square matrix of order $𝑛$ such that $A^2=A$ Prove that every $𝑣∈\mathbb R^n$ can be decomposed as $ 𝑣=𝑣_1+𝑣_2$ [closed]

Let $A$ be a square matrix of order $n$ such that $A^2 = A$. Prove that every $v \in \Bbb R^n$ can be decomposed as $v = v_1 +v_2$, where $v_1$ is in the null space of $A$ and $v_2$ is in the column ...
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Conditions for Group given Semigroup with Idempotent Element

Another exercise in Seth Warner's "Modern Algebra" (1965). This one is Exercise 7.15. Let $(S, \circ)$ be a semigroup. Let $(S, \circ)$ have an idempotent element $e$, that is, such that $e \...
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Is there general way to prove that for a given function exists formula with a constant amount of operations?

E.g there is a formula for a function that gives a number of idempotent functions for a set with finite size. The solution is $\sum_{k=1}^n{n\choose k}k^{n-k}$ but you need to sum intermediate results....
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Set of elements of an idempotent semirings are totally ordered.

An element $S$ is said to be totally ordered set if for all $a, b\in S\implies$ either $a\leq b$ or $b\leq a.$ An algebraic structure $(S, +, \cdot)$ is said to be an idempotent semiring if $x\cdot x=...
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Unit ring non-trivial where every element is idempotent [duplicate]

Let $A$ be a nontrivial unit ring such that $x^2 = x$ for all $x \in A$. Calculate $(x+y)^2$ and deduce that A is commutative. Prove that if $A$ is domain, then $A \cong \mathbb{Z}_2$. Prove that ...
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Can the image of a not-bounded "projector" on a normed space be [closed]

Let $X$ be a normed space, and $P:X\to X$ be an idempotent linear map, i.e., $P^2=P$. If $P$ is bounded, then $P(X)$ is closed. Does the converse hold? That is, if $P$ is not bounded, must $P(X)$ not ...
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Idempotency and Fixed-point combinators

In the $\lambda$-calculus, for a fixed-point combinator $P$, we have $Pf = f(Pf)$ for all functions $f$. Thus we could always expand $(Pf)$ as follows: $$Pf = f(Pf) = f(f(Pf)) = f(f(f(Pf))) = f(f(f(…f(...
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Checking primitivity of central idempotents: does it suffice to only consider central summands?

Let $R$ be a (not necessarily commutative) ring with $1$, and let $e$ be an idempotent, i.e. $e^2=e$. We call $e$ primitive if it is not the sum of two nonzero orthogonal idempotents. Assume that $e$ ...
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Is centralizing all idempotents just once enough to make a ring abelian?

A ring $R$ is called abelian if all idempotents in $R$ are central. So, every ring should have a universal abelian quotient ring. One way to do this is to construct an ascending chain of ideals $I_0 \...
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If both $k$, $k+1$ are idempotents in $\mathbb{Z}_n$ and $k≠0$ then $n=2k$

Question: If both $k$, $k+1$ are idempotents in $\mathbb{Z}_n$ and $k≠0$ then $n=2k$ My attempt: since $k$, $k+1$ are idempotents in $\mathbb{Z}_n$ Hence we have, $k^2\equiv k\mod n$ and $(k+1)^2\...
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If $\operatorname{Idem}B\rightarrow \operatorname{Idem}(B/mB)$ is surjective then $B$ is a product of local rings

Let $A$ be a local ring with maximal ideal $m$ and $B$ a finite $A$-algebra (by finite I mean that $B$ is a finitely generated $A$-module). If we denote by $\operatorname{idem}B$ (respectively $\...
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Continuous function from $[0,1]$ to set of idempotent matrices in $M_2(\mathbb{R})$

How to prove that there exists no continuous function $f:[0,1] \to \{A \in M_2(\mathbb{R})|A^2=A\}$ such that $f(0)=0$ and $f(1)=I$.
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Show that set subset of matrice of X in $M_{n}(R)$ such that X is idempotent, symmetric and with Tr(X) =1 is a submanifold

I'm trying to show that set subset of matrice of X in $M_{n}(R)$ such that X is idempotent, symmetric and with Tr(X) =1 is a submanifold of $M_{n}(R)$. My idea is to consider the set $ \{(X,(0,0,0)) \}...
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Is $\ker T = \ker T^*$ for idempotent operators?

Recall that if $T : V \to W$ is a linear map between finite-dimensional inner product spaces, then $T^* : W \to V$ is defined as the unique linear map satisfying $\langle Tv, w \rangle = \langle v, T^*...
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How can I prove that a specific symmetric matrix $P = I - \frac{uu^T}{u^T u}$ is idempotent?

Given the matrix $$P = I - \frac{uu^T}{u^T u},$$ how can I prove that $P = P^2$? I know that the matrix is symmetric, but how can I use this information to prove that it's idempotent?
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Prove a matrix is idempotent using algebra

I'd like to prove that this matrix is idempotent using a more algebraic proof for matrices with a similar definition to A, rather than deriving its eigenvalues or calculating $A^2$ . $A=$ $\begin{...
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Prove: $\det(I+A) = 2^{\text{rank}(A)}$ if $A$ is a square idempotent matrix. Find $(I+A)^{-1}$ such that the expression doesn't have inverses.

To prove: $\det(I+A)$ = $2^{\operatorname{rank}(A)}$ if $A \in$ $\mathbb{R}^{n\times n}$ and $A^{2}=A$. Find an expression for $(I+A)^{-1}$ such that it does not involve inverses. Is there any way I ...
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Jordan decomposition of Idempotent matrix.

Matrix A $\in$ $\mathbb{R}^{n\times n}$ is idempotent if $A^{2} = A$. Describe the Jordan form of A. How do I do this? I am able to decompose a matrix to its Jordan form given that the matrix contains ...
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Prove that $V=V_0\oplus V_1$ where $T(v_0)=0$ for all $v_0\in V_0$ and $T(v_1)=v_1$ for all $v_1\in V_1$, where $T$ is an idempotent linear trans.

Let $T:V\rightarrow V$ be an idempotent linear transformation. Prove that $V=V_0\oplus V_1$ where $T(v_0)=0$ for all $v_0\in V_0$ and $T(v_1)=v_1$ for all $v_1\in V_1$. EDIT: My attempt: Since $T(v_0)=...
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Characterizing an idempotent matrix

Show that if $C^-$ is the generalized inverse of a symmetric matrix $C$ such that $C\mathbf1=0$ then $\dfrac{l'C^-l}{l'l}$ is constant for all vectors $l$ such that $l'\mathbf1=0$ iff $\theta C$ is ...
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Show that all diagonal and off-diagonal entries of a matrix are same

If $\theta>0$ and $C_{n\times n}$ is a symmetric matrix of rank $n-1$ (with $\vec{\mathbf 1}$ as the only vector in kernel) such that $$\theta C^2=C$$ How to show that all the diagonal entries of $...
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Conjugate idempotents

Let $R$ be a ring, and let $1 =e_1 +\cdots+ e_r = e'_1 + ...+ e'_r$ be two decompositions of $1$ into sums of orthogonal idempotents. If $e_1\cong e_i'$ for all $i$, show that there exists $u \in U(R)$...
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2 answers
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On an example of an idempotent which is not a projection

I am working through a solution of the following exercise in Conway's functional analysis: Let $\mathscr{H}$ be the two-dimensional real Hilbert space $\mathbb{R}^2$, $\mathscr{M}=\{(x,0):x\in\mathbb{...
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1 answer
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$\operatorname{trace}(e_1e_2)=\operatorname{trace}(f_1f_2)$ for certain idempotents of a $C^*$ algebra with trace

Indirectly inspired by this post we ask the following question: Let $A$ be a $C^*$ algebra which is equiped with a faithful positive normal trace. Assume that $e_1,e_2,f_1,f_2$ are idempotents in $...
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About artinian rings and homomorphisms

Let $R$ be a left artinian ring, $\mathfrak{r}:=J(R)$. and e,f idempotent in $R$. Proof that the morphism of abelian groups $\varphi:eRf\rightarrow Hom_R(Re,Rf)$, $\varphi(erf)(r'e):=r'erf$ is an ...
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A question about v-prehomomorphisms of inverse semigroups

In this paper D. B. McAlister introduced a very interesting class of morphisms for inverse semigroups, which he called v-prehomomorphisms. For a such morphism $\theta : S \to T$ we have $\theta(ab)\...
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$R$ such that $R/J$ is artinian ($J$ is the Jacobson radical) and there exists an idempotent cannot be lifted modulo $J$?

I know a few examples of rings and ideals such that there exists an idempotent cannot be lifted modulo the ideal (for instance, $\mathbb Z$ and $n\mathbb Z$). My question is: is there a ring $R$ such ...
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$R$ is semiperfect and $e^2=e\in R$ $\Longrightarrow$ $eRe$ and any factor-ring of $R$ is semiperfect

Let $R$ be a semiperfect ring and $e^2=e\in R$ an idempotent. Prove that $eRe$ and any factor-ring of $R$ is semiperfect too. I know that $R$ is a semiperfect ring iff $R$ decomposes to a direct sum ...
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The left annihilator of idempotent elements

$R$ is a ring with identity.If $\ \pi\in R\ $ is an idempotent element ($\pi^2=\pi$),How to prove $\operatorname{ann}_\text{l}(\pi)=R(1-\pi)$? where $\operatorname{ann}_\text{l}(\pi)$ represents the ...
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Proving that $(2k+1)$ is idempotent for $\mathbb{Z}_{2(2k+1)}$ [duplicate]

I've noticed that the group $\mathbb{Z_n}$ under multiplication has a non-trivial idempotent element particularly when $n=2(2k+1), k \in \mathbb{N}$ --- in other words, the even numbers are twice an ...
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2 votes
1 answer
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Let $T$ be a linear operator such that it is both self-adjoint and unitary

Let $T$ be a linear operator on a finite dimensional inner product space $V$ such that it is both self-adjoint and unitary.Then prove that it can only arise from the subspace $W$ s.t. $$V=W \bigoplus ...
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3 votes
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If $A$, $B$ idempotent and $AB=0$, then $A+B$ idempotent.

We know that if $A,B$ idempotent then we have (see edit) $$(A+B)^2=A+B\implies AB=0,$$ and I'm wondering whether the converse, i.e. that if $A,B$ idempotent and $AB=0$ then $A+B$ idempotent is true. ...
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Generalisation of idempotent matrices to higher powers

Does there exist a species of matrix, say $M$, such that $$M^{\alpha} = M, \quad M \neq I$$ for some $\alpha > 2$. I'm not looking for idempotent matrices (square to themselves), but in essence a ...
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1 answer
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How do I show that if the product of idempotents is idempotent, then idempotents commute with other elements?

I was not able to prove the following theorem. I would appreciate some help regarding it: If the product of each two idempotent elements in a unity ring is an idempotent element itself, prove that ...
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1 answer
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Show that there exists a basis

Let $g$ be an idempotent endomorphism. Show that there exists a basis $B$ for $V$ such that for some $v_i\in B$: $g(v_i) = v_i$ for $1 \leq i \leq r$ and $g(v_i) = 0$ for $r+1\leq i \leq n$, where $r =...
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1 vote
3 answers
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Proving idempotent elements in $\mathbb{Z}_n$

I'm working in $\mathbb{Z}_n$ under multiplication (not addition) for some time now and I'm beginning to notice something: For $n \in \mathbb{N}$ where $n$ is even and $\frac{n}{2}$ is odd, it seems ...
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1 vote
3 answers
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Why is $I$ the only idempotent matrix with nonzero determinant?

While reading about the attribute of the identity Matrix, it's mentioned that I is not only idempotent but that it is also the only such matrix that does not have a determinant of zero. While I being ...
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2 votes
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Idempotent elements of $Z_4 \times Z_8$

I am trying to find idempotent elements of the direct product $Z_4 \times Z_8$. By the definition of direct product we have: $$Z_4 \times Z_8=\left\{(0,0),(0,1),.....(2,7),(3,7)\right\}$$ Now by the ...
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1 vote
1 answer
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Product of non-idempotent elements in a commutative monoid

Assuming we have a commutative monoid $(M,\cdot)$ such that the non-trivial elements have no inverse. In addition, M contains no non-trivial idempotents. Considering two non-trivial elements $a_1$ and ...
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1 answer
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Idempotent elements in a C^*- algebra

While reading the proposition above, I'm stuck on understanding the three parts below. Why is $z$ invertible? How did we get $ez= ee^*e$? Why is $(1-te+tp)$ invertible? Thank you in advance!
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$\underset{K}{\operatorname{arg min}} [tr(KDK^TA)-tr(K^TA)]$ when $K$ is idempotent and $D$ diagonal?

I am trying to find the minimum $\underset{K}{\operatorname{arg min}} [tr(KDK^TA)-tr(K^TA)]$ where $K$ is an idempotent matrix, $D$ a diagonal matrix and $A$ a positive semidefinite matrix ($K, D, A \...
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1 vote
1 answer
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Idempotent endomorphism: show $M=\text{Range}(f) \oplus \text{Range}(\text{id} - f)$ [duplicate]

let $f$ be in idempotent endomorphism. Show that $M=\text{Range}(f) \oplus \text{Range}(\text{id} - f)$. I want to show the intersection of both range spaces is $\{0\}$. I currently have let $v \in \...
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6 votes
1 answer
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Why is the linear Span of the idempotents in a Banach algebra a Lie ideal?

In https://arxiv.org/pdf/1505.04503.pdf it was mentioned that "the linear span of the idempotents is a lie ideal". I wonder where I can find a proof for that. Let $A$ be a Banach algebra and ...
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