# Questions tagged [idempotents]

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

275 questions
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### 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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### 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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### 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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### 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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### How do you prove there is a bijection between an isomorphism and a set of orthogonal idempotents?

Let $A$ be a commutative ring with unity. How do you prove there is a bijection between: An isomorphism as a product of rings $\phi: A \longrightarrow A_1\times\cdots\times A_n$ A decomposition as ...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### Is $A^3=A$ a condition for idempotency of matrices?

Given that $A$ and $B$ are two idempotent (square) matrices of same order, $AB+BA=AB-BA=O$ (Where $O$ is the null matrix of the same order). Prove that both $A+B$ and $A-B$ are idempotent. I ...
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### Composition series of regular module

Assuming I know all (primitive) idempotents of an algebra $A$, and its centre, is there a way to obtain a composition series for the regular module $_A A$, and the simple modules of $A$?
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### Relations of structures related to conjugate idempotents

If $e$ and $f$ are conjugate idempotents in some algebra $A$, I guess the modules $Ae$ and $Af$ should be isomorphic, as well as the algebras $eAe$ and $fAf$ . Are the maps canonically given by just ...
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