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.
...
0
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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 ...
6
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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.
$$(I - P)^2 = I^2 - IP - PI + P^2 = I - ...
3
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1answer
91 views
How to prove this rank inequality?
Let $n\geq2$ and $A,B\in M_{n}(\mathbb{C})$ such that $B^2=B$. Prove that $$\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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votes
2answers
82 views
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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1answer
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.
...
3
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1answer
26 views
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 ...
2
votes
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?
1
vote
0answers
77 views
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
87 views
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 \...
1
vote
4answers
95 views
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 ...
6
votes
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 $J$, ...
0
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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
43 views
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
145 views
$A$ 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
20 views
$R$ 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
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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
58 views
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 ...
1
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1answer
27 views
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($f,g$) ...
3
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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 $f:\mathbb{Z}\times \mathbb{Z}\longrightarrow \mathbb{Z}\times \mathbb{Z}$ ...
2
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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.
1
vote
0answers
25 views
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 $Re+Rf=Rg$
1
vote
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?
2
votes
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....
1
vote
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 -...
1
vote
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 ...
