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Questions tagged [idempotents]

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

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Relationship of the trace between a matrix multiplied by its logarithm

I was indecisive about whether to post this problem in the Physics forum or in the Mathematics one. However, since I am mostly interested in the mathematical understanding of it, I am posting it here. ...
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1answer
40 views

Let $V$ be a real vector space and $E$ be an idempotent linear operator on $V$. Prove that $I + E$ is invertible.

Let $V$ be a real vector space and $E$ be an idempotent linear operator on $V$, that is a projection. Prove that $I + E$ is invertible. Find $(I + E) ^{-1}$ My teacher taught me the following proof ...
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1answer
116 views

Show that $(I − P)^2 = I − P$ if $P=P^2$

Let $P $ be an $n \times n$ matrix and $I$ be the $n \times n$ identity matrix. Show that $$ (I − P)^2 = I − P $$ is valid if $P = P^2$. I did the following. $$(I - P)^2 = I^2 - IP - PI + P^2 = I - ...
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1answer
91 views

How to prove this rank inequality?

Let $n\geq2$ and $A,B\in M_{n}(\mathbb{C})$ such that $B^2=B$. Prove that $$\mbox{ rank }(AB-BA)\leq\mbox{ rank }(AB+BA).$$ If $B$ is zero or the identity matrix, we are done. But $B$ will always be ...
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1answer
57 views

Why does $em=(1-e)m$ imply $m=0$?

Let $R$ be a ring and $M$ be a module over it. Suppose $e\in R$ is an idempotent and $m\in M$. Why does $em=(1-e)m$ imply $m=0$? I can't see how I can use the property of idempotents other than ...
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2answers
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Why does $Re=S^{-1}R$?

The problem is to show that if $e$ is an idempotent in a ring $R$, then $Re=S^{-1}R$ where $S=\{1,e,e^2,e^3,\dots\}=\{1,e\}$. In fact this doesn't even seem plausible to me, because $Re$ is "smaller" ...
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2answers
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Is this modest change enough to make an idempotent element an identity element?

In The Number System by Thurston the author introduces an algebraic structure he calls a hemigroup. The laws of a hemigroup are: (i) $\left(x\odot y\right)\odot z=x\odot \left(y\odot z\right),$ (...
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1answer
30 views

Find the Jordan Canonical Form that is similar with the idempotent matrix A

Find the Jordan Canonical Form that is similar to the idempotent matrix $A$. I know that since $A=A^2$ then $A(A-I)=0$ so the minimal polynomial is $m_A(\lambda)=\lambda(\lambda-1)$. I also know ...
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0answers
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Orthogonal idempotent, Dunkl operators

I would like to check te following: Let $V$ be a $\mathbb{C}$-vector space of dimension ($n$); let $G$ be a complex reflection group (see BMR); let $\mathcal{A}$ the hyperplan arrangemant of ...
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2answers
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Existence of an idempotent element $\not = 1$ and $\not = 0$

I have a problem to solve the following problem: If $1=e_1+e_2$ with non-units $e_1,e_2 \in R$ and if $e_1e_2$ is nilpotent, then there is an idempotent element $e\not =0$,$e\not =1$. Perhaps it is a ...
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1answer
30 views

Generating idempotent (cyclic code length 7)

I have given a cyclic code $C$ generated by $g(x) = x^3 + x + 1$. Now, I'm looking for the generating idempotent of $C$. Is it correct that I have to find the factorization of $x^3+x+1$ over GF(2), ...
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0answers
50 views

Prove that there's only one matrix which is involutory and idempotent at the same time.

An involutory matrix $\mathbf{A}$ is such that $\mathbf{A}^{2}=\mathbf{I}$. An idempotent matrix $\mathbf{A}$ is such that $\mathbf{A}^{2}=\mathbf{A}$ So we'd have to satisfy both conditions ...
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2answers
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All idempotents splits in set theory context

So I wanted to show that in the category $\mathbf{SET}$, all idempotent splits. Fix an idempotent $f$, I wish to find 2 arrows $g,h$ such that $f=hg$ and $gh = 1$. I'm having some trouble "...
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1answer
66 views

Determine if a matrix is an orthogonal projection matrix

We define an orthogonal projection as a linear transformation that maps a vector into its orthogonal projection in some (given ahead) subspace $W$. Let's call the matrix of that transformation (...
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1answer
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Given P idempotent, show that I−P is idempotent. [duplicate]

So my task is well summed up by this older post: Given $P$ idempotent, show that $I-P$ is idempotent. PandaMan idea is that by proving $(I−P)^2 = (I-P)$ we prove that $(I-P)$ which implies that ...
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2answers
72 views

Idempotents in $ \mathbb{Z}_n $ [closed]

Let $ n=cd $ where $ c $ and $ d $ are co-primes. Then there are integers $ x $ and $ y $ such that $ xc+dy=1 $. How can it be proved that $ xc$ is idempotent in $ \mathbb{Z}_n $? Is converse ...
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4answers
144 views

Show that if A = B + C, then BC = 0

A,B,C are n $\times$ n symmetric and idempotent matrices. The question is: If $$ \textbf{A = B + C } $$ then show $$ \textbf{BC = 0 } $$ I'm not sure where to start?
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1answer
31 views

matrix algebra and idempotent matrix

I'm having a little trouble understanding a few derivations in my book for least squares regression. $\textbf{Question 1}$: If $\textbf{M}^0 \textbf{i} = [\textbf{I} - \frac{1}{n}\textbf{ii'}]\...
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1answer
79 views

Find the number of elements in a quotient ring satisfying $a^2 = a$

Let $f(T)=T^3+T^2+2T+2 \in \mathbb{Z}[T]$ and let $I$ be the ideal of the ring $\mathbb{Z}[T]$ generated by $f(T)$ and $5$. Find the number of elements $z \in \mathbb{Z}[T]/I$ such that $z^2=z$. Some ...
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1answer
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Is there a name for an associative algebraic structure in which everything is irreducible?

Let $A$ be a set and $\ast$ a binary operator on that set. Let us suppose that $(A,\ast)$ satisfies the following axioms: For all $x,y,z \in A$, $x \ast (y \ast z) = (x \ast y) \ast z$ For all $x,y,z ...
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53 views

Relation between simple roots and idempotents of quotient ring

Given a ring $A$ and a polynomial $p\in A[x]$, write $Z(p,A)$ for the simple roots in $A$ of $p\in A[x]$. On the other hand, consider the set $\mathrm{Idemp}(A[x]/(p))$ of idempotents of the quotient. ...
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1answer
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lifting orthogonal idempotents (induction step)

I'm trying to prove (by induction on $n$) that whenever $I$ is a nilpotent ideal of a ring $R$, and $r_1+I,\ldots,r_n+I$ form a complete set of orthogonal idempotents in the quotient $R/I$, there ...
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1answer
156 views

If $a^2=a$ find the inverse of $1+a$ [closed]

If $a^2=a$ find the inverse of $1+a$. Note that $a\in R$ where $R$ is a ring and inverse of $1+a$ exists in $R$ I tried $(1-a)(1+a)=1-a^2=1-a$ But I need $1$ in the RHS.How to get it?
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0answers
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Proving that $\mathbf{(H-\frac{1}{n}J_n)}$ is indempotent

I am trying to show that the matrix $\mathbf{(H-\frac{1}{n}J_n)}$ is idempotent where $\mathbf{H}$ is the Hat-matrix (Projection matrix) of linear regression and $J_n$ is the $n\times n$ matrix with $...
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0answers
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Idempotents and number of proper cyclic codes

So, there was a question on my exam last year as follows: Using idempotents, determine the number of proper cyclic codes of length 17. I got the question right, but I can't remember how to do the ...
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1answer
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GRE 9768 #60 1. Does $(s+t)^2=s^2+t^2$ imply $s+s=0$? 2. Idempotent matrices do not form a ring?

GRE 9768 #60 on what appears to be Boolean rings: Ian Coley's approach is to prove $(I)$ and $(I) \implies (II) \implies (III)$ I think $(II) \implies (I)$. My attempt: $$(s+t)^2=s^2+t^2 \...
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4answers
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Finding the idempotent matrix

I am faced with the problem: Let $p$ $=$ $\begin{pmatrix} 1\\2 \end{pmatrix}.$ Find the idempotent matrix M such that $Mv$ is orthogonal to $p$ for any $2 \times 1$ vector $v$. I understand that ...
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1answer
95 views

Question about the transpose of identity and hat matrices

So I know that because the hat matrix and identity matrix are both symmetric, $H^{T}=H$ and $I^{T}=I$, respectively so would $(I-H)^{T}=(I-H)$? Sorry if the answer is obvious, I am missing a step ...
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2answers
58 views

What property of a function describes $f\left(f\left(f\left(\phi, x\right), y\right), z\right) == f\left(\phi, z\right)$?

How would you describe a function that has the relationship $f\left(f\left(f\left(\phi, x\right), y\right), z\right) == f\left(\phi, z\right)$? given that $f\left(\phi, x\right)$ is idempotent (i.e., $...
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1answer
68 views

How to dram Hasse Diagram in a lattice theory

Let $E$ be the set of all nonzero idempotents from the commutative ring $\mathcal{A}$. An atom in $E$ is an element $e$ in that set such that $ef = f$ for some $f \in E$ implies $f = e$. In other ...
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2answers
202 views

Peirce decomposition of a ring: must the ideal generators be idempotent in characteristic 2?

I'm considering this particular statement of the Peirce decomposition of a ring: If a commutative unital ring $R$ can be written as an internal direct sum of two of it's proper ideals $I$ and $J$, ...
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1answer
48 views

Understanding the linear transformation given by matrix multiplication

I am working on the following problem. I found it in an old qualifying exam, but I'm not sure of its original source. It asks: Let $A$ be an $n \times n$ matrix with entries in $\mathbb{R}$. Let $\...
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0answers
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Transpositions and idempotents in matrix algebras

Let $A$ be a semi-simple algebra over $\mathbb{C}$. An idempotent $x$ in $A$ is an element of $A$ such that $x^2 = x$. A minimal non-zero idempotent $y$ of $A$ is a non-zero idempotent of $A$ such ...
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5answers
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$A$ is a symmetric matrix such that $A^4=A$. Prove that $A$ is idempotent

Let $A$ be a real symmetric matrix such that $A^4=A$. Prove that $A$ is idempotent. I have tried using eigenvalues and only inferred that the eigen values may be $0,1$. But I cannot proceed with this....
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1answer
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$R$ is artinian $\implies$ $(eR)_R$ is local $R$-module for an idempotent $e \in R$

Let $R$ be a ring with unity $1$ and artinian. Consider $R$ as right $R$-module $R_R$. Since it is artinian we have a finite decomposition of $R$ into right ideals: $$R_R = A_1 \oplus ... \oplus A_n$$...
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1answer
79 views

A characterization for idempotent lifting property

Let $I$ be an ideal in a commutative ring $R$ with $1$ and let $g+I$ be an idempotent element of $R/I$. We say that this idempotent can be lifted modulo $I$ in case there is an idempotent $e^2=e\in R$...
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1answer
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Homotopy idempotent maps over a finite product of spheres

Let $X$ be a topological space. A continuous map $h:X\longrightarrow X$ is called homotopy idempotent if $h\circ h\simeq h$. My question is: What is the number of homotopy classes of homotopy ...
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1answer
50 views

Decomposition of idempotent matrix

Let be $P\in \mathbb{R}^{n\times n}$ of rank $r\leq n$ and idempotent, i.e. $PP=P$. I want to show that there exist matrices $A\in \mathbb{R}^{n\times r}$ and $B\in \mathbb{R}^{r\times n}$ such that $...
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1answer
123 views

Lifting idempotents modulo an ideal and its radical

The following result is a direct consequence of Proposition 27.1 of F. W. Anderson and K .R. Fuller's "Rings and categories of modules". But I cannot prove it. Is there any hint? Definition: Let $A$ ...
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1answer
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Need help in proving $P$ is a projection.

A bounded linear operator $P:\mathbb H\rightarrow\mathbb H$ on a Hilbert space $\mathbb H$ is a projection iff $P$ is Self-adjoint and idempotent Proof:Initially,i started by assuming $P$ to be a ...
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1answer
36 views

Can I use Idempotency Here?

If I have statement like: AB + 'AC + BC, can I use Idepotency to remove BC and simplify, or does the AND between AB and 'AC rules this out? More specifically to simplify: AB+BC+C'A - Commutativty ...
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0answers
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Primitive idempotents of Cl(1,3) over the complex numbers

Simply put, I need to find all primitive idempotents of the Clifford Algebra $Cl(1,3)$ over the complex numbers. I have found some general results but they're only applicable over the real numbers. I ...
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1answer
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Idempotents and Factorization

In the ring $\mathbb{F}_2[x]$, let $f(x)$ be a reducible polynomial of degree $n$. If we happen to know that there is a non-trivial idempotent $g$ in $\mathbb{F}_2[x] / (f(x))$, then one of gcd($f,g$) ...
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3answers
110 views

Idempotent endomorphisms on $\mathbb{Z}\times \mathbb{Z}$ [closed]

Recall that a homomorphism $f:G\longrightarrow G$ is called idempotent if $f\circ f=f$. What are idempotent homomorphisms $f:\mathbb{Z}\times \mathbb{Z}\longrightarrow \mathbb{Z}\times \mathbb{Z}$ ...
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2answers
229 views

Do you know if eigenvectors of idempotent matrices are always orthonormal?

I'm currently studying Econometrics theory, and I'm stuck in my problem set where it is asked if "eigenvectors of idempotent matrices are always orthonormal". How can I justify that? Cheers.
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0answers
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Regular Ring & principal ideals of idempotents [closed]

If $R$ be Regular Ring. And for any two idempotent elements $e$ & $f$ there exist a idempotent element $g$ such that $Re+Rf=Rg$
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1answer
323 views

Must an idempotent matrix be symmetric?

A matrix $A$ is idempotent if: $$AA = A$$ Is it true that all such matrices are symmetric?
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1answer
41 views

Proving lemma about centrality of idempotent elements in a Ring with no nilpotent elements.

Reading the class notes I stumbled upon an unproved lemma that I am having issues proving. The lemma is, if $R$ is a ring with no non-zero nilpotent elements and $e$ is idempotent then $e$ is central....
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1answer
31 views

Idempotents over a ring with zero divisors

Let $R = \mathbb{Z}/4\mathbb{Z} = \{0, 1, 2, 3\}$ and the group $G = \mathbb{Z}/2\mathbb{Z} = \{e, a\}$. Consider the group ring $R[G]$. I have read somewhere (1) that $\frac{e + a}{2}$ and $\frac{e -...
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1answer
96 views

Is it true that $X(X'X)^{-1}X'-J/n$ is idempotent, where $J$ is an $n$ by $n$ matrix of ones?

$X$ is a full column rank $n$ by $p$ matrix with the first column a vector of ones. Now the I was trying to prove, from a different approach that the SSR/variance is Chi square but this means I have ...