All Questions
6,548 questions
2
votes
2
answers
617
views
Comparing lower central series and augmentation ideal completions
Let G be a group. Let $G^p$ be the completion of G with respect to the mod p lower central series of G.i.e. $G^p=\varprojlim_{q} G/\gamma_qG$, where $\gamma_qG$ is generated by all $\{[x_1,\cdots,x_s]^...
- 22.6k
19
votes
3
answers
2k
views
Is the semigroup of nonnegative integer matrices with determinant 1 finitely generated?
The group of $n\times n$ matrices with integer entries and determinant equal to 1, $SL(n,Z)$, is a finitely generated group (in fact, it is generated by 2 matrices). I am interested to know if the ...
- 25.1k
1
vote
2
answers
660
views
Markov chain convergence problem.
Consider a markov chain matrix P of size n x n (n states).
P is known to be:
1- there are at least two absorbent states. one of them is denoted by null. (thus, we have that P_null,null = 1)
2- For ...
CommunityBot
- 1
13
votes
1
answer
1k
views
Why not _co_free modules?
Let $R$ be a ring, and $R\text{-Mod}$ its category of all left modules. There is a "forgetful" functor $\operatorname{Forget}: R\text{-Mod} \to \text{AbGp}$, which is additive, continuous, and ...
- 57.6k
9
votes
2
answers
2k
views
What are examples of cogenerators in R-mod?
Fill in the blank, please :)
Let $\mathcal C$ be a complete and cocomplete abelian category. A generator in $\mathcal C$ is an object $X \in \mathcal C$ such that every object $Y \in \mathcal C$ ...
- 15.6k
1
vote
2
answers
441
views
Quartic form which is irreducible but not geometrically irreducible
Suppose that $F$ is a finite field of odd characteristic.
Suppose that $q(X_1,X_2,...,X_n)$ is a quartic (homogeneous) form with coefficients in $F$ such that
$q$ is irreducible over $F$
$q$ does ...
- 3,137
10
votes
3
answers
1k
views
Alternating forms as skew-symmetric tensors: some inconsistency?
My trouble is best described by the following diagram:
$$ \begin{array}{ccccc} \mathrm{Alt}^k V &\stackrel{\sim}{\rightarrow}& (\Lambda^k V)^* &\stackrel{\sim}{\rightarrow}& \Lambda^...
- 422
5
votes
1
answer
706
views
(Stochastic) matrix for which a stochastic matrix logarithm exists?
I think this is basically the inverse question of Matrices whose exponential is stochastic.
i.e. what are sufficient conditions on the matrix representation of an evolution operator of a (finite) ...
CommunityBot
- 1
7
votes
3
answers
744
views
Looking for applications of a nice result in linear algebra
Hello everybody
There is a nice classical result in linear algebra: if $A, B$ are two matrices in $M_n(k),$ where $k$ is a field, and $B$ commutes with every element of $M_n(k)$ which commutes with $...
- 14.5k
7
votes
1
answer
372
views
Simultaneously orthogonally transform two SPD matrices to tridiagonal form?
Supposing you have two SPD matrices $A,B\in\mathbb{R}^{n\times n}$ are there any known results on the existence or non-existence of a unitary matrix $Q$ such that $Q^\top A Q=T_A$ and $Q^\top B Q=T_B$ ...
1
vote
0
answers
2k
views
Generators of ideals in polynomial rings over commutative rings.
This is my first question; I hope it worthy of this awesome forum and its members.
Let $R$ be a commutative ring, perhaps with unit, perhaps not. As usual let $R[x]$
denote the ring of polynomials ...
- 391
3
votes
0
answers
614
views
nilpotent matrices over polynomial rings
I am looking for an analogue of the Jordan normal form for nilpotent matrices over the
polynomial ring ${\mathbb Z}[x_1, \dots, x_n]$. More precisely, is there a description for the orbits of action ...
- 6,214
4
votes
2
answers
413
views
Universal functors according to Cohn.
In section III.1 of P.M. Cohn's Universal Algebra a notion of universal functor ${\cal L} \rightarrow {\cal K}$ is defined for a representation of one category in another given by a (covariant) ...
- 73.1k
8
votes
2
answers
1k
views
Criterion for an abelian group to have a commutative endomorphism ring
Given an abelian group $G$, one can form the endomorphism ring $\mbox{End}(G)$ by letting $\alpha+\beta=\alpha(x)+\beta(x)$, and $\alpha\beta=\alpha(\beta(x))$, where $\alpha$ and $\beta$ are ...
- 8,467
1
vote
1
answer
201
views
How can I characterize the type of solution vector that comes out of a matrix?
Ax = b. I need a way to analyze a square matrix A to see if its solution vector x will ...
- 156k
9
votes
1
answer
679
views
Mathematical software for computing in integral group rings of discrete groups?
I'm doing computations in the integral group ring of a discrete group,
in particular the discrete Heisenberg group. In this case elements
are integral combinations of monomials $x^k y^m z^n$, where ...
- 5,654
6
votes
1
answer
317
views
sheaves of modules on an $\ell$-space
Let $X$ be a Hausdorff, locally compact, and totally disconnected topological space, which I call an $\ell$-space, and write $A = C^{\infty}_C(X)$ for the algebra of locally constant complex-valued ...
- 7,273
4
votes
2
answers
1k
views
Representation of rings
The endomorphisms of an abelian group form a ring under pointwise group operation and composition. Every ring is isomorphic to a subring of the endomorphism ring of some abelian group (left module ...
- 7,442
3
votes
2
answers
1k
views
endomorphism ring of a finite-length module
Can anyone tell me why the endomorphism ring of a finite-length module is artinian? Bonus points if you can do it without using the radical, semisimplicity, Fitting's lemma or anything fancy. If you ...
- 1,198
1
vote
0
answers
576
views
Minimizing quadratic form over permutations
Let $Q$ be an $n \times n$ real symmetric matrix and $x$ an $n \times 1$ real vector. Consider the following minimization problem:
$\min_{\pi \in S_n} ~(\pi x)^{\rm T} Q (\pi x)$,
where $S_n$ ...
- 1,839
7
votes
3
answers
3k
views
terminology about ring/algebra in abstract algebra and measure theory
Both in abstract algebra and measure theory is there term ring/algebra, but their definition are different and we can not deduce one from the other, the only requirement in definition they share is ...
CommunityBot
- 1
2
votes
2
answers
378
views
Efficient computation of AB^-1 for matrices
Hi there,
Sorry if this has already been asked before. I tried googling for it, but perhaps I could not find the right words to search for. My question is: Which is the fastest way to compute A*inv(B)...
3
votes
2
answers
2k
views
Extension problem
As I understand, if $0\rightarrow A\rightarrow X\rightarrow B\rightarrow 0$ is a short exact sequence of abelian groups, $\mbox{Ext }_{\mathbb{Z}}^{1}(B,A)$ gives all the isomorphism classes of what ...
- 33.8k
4
votes
1
answer
1k
views
Stability of Conjugate Gradient Method
When dealing with spd matrices of relatively low condition number, how likely is it (and is it easily provable) that the conjugate gradient method will always be able to find the solution without ...
- 20.2k
5
votes
2
answers
2k
views
Matrices whose exponential is stochastic
The complex matrix exponential of a Hermitian matrix is unitary: $e^{-iH} = U$. Is there a name or a characterization for matrices Q whose real exponential is stochastic: $e^{-Q} = S$?
- 60.5k
20
votes
1
answer
3k
views
On a theorem of Jacobson
In a comment to an answer to a MO question, in which Bill Dubuque mentioned Jacobson's theorem stating that a ring in which $X^n=X$ is an identity is commutative (theorem which has shown up on MO ...
CommunityBot
- 1
4
votes
0
answers
179
views
Global dimensions of orders over non-Gorenstein centers
This question concerns the following Lemma 4.2 in this paper by Van den Bergh:
Lemma: Let $R$ be local, normal Gorenstein ring of dimension $d$. Suppose $M$ is a reflexive $R$ module such that $A=\...
- 30.5k
3
votes
2
answers
170
views
Equivariant maps of "higher order"
Given a group $G$, a ring $R$ and two $R[G]$-modules $M,N$. Then one can consider $Hom_R(M,N)$ and define inductively submodules $A_0,A_1,...$ via
$A_0:=0$
$A_{n+1}:=\{ \;f\; |\; \forall\; g\in G: (...
- 12.6k
28
votes
4
answers
5k
views
When are modules and representations not the same thing?
I've been trying for a while to get a real concrete handle on the relationship between representations and modules. To frame the question, I'll put here the standard situation I have in mind:
A ring $...
- 14.5k
1
vote
1
answer
2k
views
Cardinality of symmetric group [duplicate]
Possible Duplicate:
Cardinality of the permutations of an infinite set
Why does the symmetric group on an infinite set X have the cardinality of the power set ${\cal P}(X)$?
CommunityBot
- 1
1
vote
0
answers
533
views
Integral element in the quotient of a polynomial ring
Hello,
I'm writting a "report" (to learn) on algebraic geometry, and was looking to write a proof for the following statement :
Theorem : Let $K$ be an algebraically closed field and $C_1, C_2 \...
- 11
2
votes
1
answer
1k
views
For which rings does there exist an invertible Vandermonde matrix?
Suppose $R$ is a commutative ring, and $S \subset R^{n\times n}$ is an $R$-module. We are given $H_0,\dots,H_n \in R^{n\times n}$, and we know that for all $r \in R$,
$$H_0 + r H_1 + \dots r^n H_n \in ...
- 55.9k
5
votes
4
answers
388
views
Familiar equations in more general settings
What equations, or results about equations, generalize in interesting ways from number theory or geometry to more abstract settings? The motivating example for this question was as follows:
...
- 5,217
6
votes
3
answers
3k
views
minimize the sum of absolute eigenvalues
Hi,
We have a real symmetric matrix M with diagonal elements 0's, the eigenvalues and eigenvectors of M are computed.
Now we wish to change its diagonal elements arbitrarily to minimize the sum of ...
- 817
3
votes
3
answers
3k
views
Infinite hermitian matrix
Suppose we have a finite square n x n matrix of complex numbers H that is Hermitian and skew-symmetric:
$H^\dagger = H$ and $H^T = -H$.
(T denotes transpose, $\dagger$ denote conjugate transpose. I ...
- 1,471
4
votes
1
answer
496
views
Is there a standard measure for how close a matrix is to being a distance metric ?
Suppose I have a square n*n, symmetric matrix with positive elements and zero diagonal.
For this to be considered a proper distance metric between n points, the triangle inequality needs to be ...
- 4,462
1
vote
2
answers
652
views
Understanding the modules of semiprimitive rings
As far as I understand, a semiprimitive ring can be fully 'explored' by its simple modules, in the sense that a semiprimitive ring is the subdirect product of its simple modules (for brevity, I'll use ...
CommunityBot
- 1
3
votes
2
answers
518
views
Explicit representations of finite fields
An old question that occurred to me again recently: are there any explicit formula known for sequences of irreducible polynomials $g_{p^n}(X)$ in $Z/pZ[X]$ such that for the finite field with $p^n$ ...
- 12.6k
6
votes
2
answers
1k
views
Are the banded versions of a positive definite matrix positive definite?
Consider $M$, a positive definite matrix. Let $M^{(1)}$ be the diagonal matrix which agrees with $M$ on the diagonal ($M_{ii}=M^{(1)}_{ii}$). We have that $M^{(1)}$ is positive definite because it is ...
CommunityBot
- 1
1
vote
0
answers
268
views
Rational map defined over K leads to algebra question
Hello,
Concrete algebraic question : Let $K$ be a perfect field, $\bar{K}$ a fixed algebraic closure and let $f \in \bar{K}[x_1,\ldots,x_n]$. I was wondering when there exists another polynomial (non-...
- 11
1
vote
1
answer
978
views
What is the term for a matrix with spectral radius less than 1, with all eigenvalues of modulus = 1 associated with a 1x1 Jordan block?
I am looking at the matrices described in the title: matrices where all eigenvalues lie in the unit disc, and with the eigenvalues of modulus 1 having 1x1 Jordan blocks. My question is, is there a ...
- 3,343
8
votes
1
answer
459
views
Semisimple-ish rings!
Let S be the class of all rings R which have 1 and satisfy this condition:
for every "non-zero" right ideal I of R there exists a "proper" right ideal J of R such that I + J = R. (The + here is not ...
- 6,734
2
votes
0
answers
4k
views
Eigenvalues of sum of commuting matrices [closed]
With reference to the following thread :
Eigenvalues of Matrix Sums
Answer by Jonas Meyer is as follows :
If 2 positive matrices commute, than each eigenvalue of the sum is a sum of eigenvalues of ...
CommunityBot
- 1
1
vote
1
answer
736
views
Matrix Conjugates over Finite Fields
Thinking about Diffe-Hillman for matrices brought me to the following question.
Given $\mathbb{F}_{p^k}$ the finite field with $p^k$ elements when can we find non-trivial solutions to
$\begin{...
- 20.8k
7
votes
2
answers
1k
views
A question on curved algebras, papers by Positselski and E. Segal
I am trying to understand something about curved dg algebras as studied by Positselski, E. Segal. These come up in mirror symmetry and when one wants to study Kozsul duality for algebras that are more ...
- 15.6k
4
votes
2
answers
920
views
What is the correct formulation of the CDE triangle?
The CDE triangle with C Cartan matrix and D decomposition matrix is well-known for finite groups. I have not seen a full account of this for finite dimensional algebras which I find surprising as ...
- 44.7k
3
votes
1
answer
350
views
Special subalgebras of central simple algebras
In this question F is a field and all algebras are finite dimensional F algebras.
Let X be the set of all F algebras A for which there exist an F algebra B and an F division algebra D such that F is ...
- 20.8k
3
votes
2
answers
375
views
Invariant subspaces of subalgebras of $M_n(C)$
Given a subalgebra E of $M_n$ (nxn complex valued matrices), what can we say about the subspaces F of $M_n$ such that $EF \subset F$? Googling for an answer gives me the reference:
Israel Gohberg, ...
- 18.7k
2
votes
2
answers
295
views
References for traceless and/or imaginary Octonionic matrices?
Hi all.
I was wondering if anyone has seen any work related to either traceless matrices of Octonions (with trace defined as the sum of diagonal) or matrices of pure imaginary Octonions (meaning real ...
- 8,597
8
votes
1
answer
1k
views
Example of an algebra finite over a commutative subalgebra with infinite dimensional simple modules
Let $A$ be an algebra over an algebraically closed field $k.$ Recall that if $A$ is
a finitely generated module over its center, and if its center is a finitely generated
algebra over $k,$ then by the ...
- 12.3k