All Questions
6,548 questions
5
votes
2
answers
680
views
Finding the solution to b = Ax that minimizes the Hamming weight (everything over the field F_2).
Is there an efficient algorithm for finding the solution $x$ of
$b = Ax$
that minimizes the Hamming weight of $x$, where
$A$ is a nxm-matrix over the field $\mathbb{F}_2$ ("integer matrix modulo 2")...
- 17.9k
8
votes
1
answer
570
views
Is there an interpretation of the "anticommutative" symmetric monoidal structure on $\mathbb{Z}$-graded abelian groups in terms of $\mathbb{G}_m$ actions?
The category of $\mathbb{Z}$-graded abelian groups is equivalent to the category of comodules over the commutative Hopf algebra (over $\mathbb{Z}$) $A=\mathbb{Z}[t,t^{-1}]$, with comultiplication $t\...
- 54.6k
0
votes
0
answers
168
views
Working in a ring with something similar to elliptic curve factorization?
Elliptic curve factorization tries to factor integer $n=pq$ by working on an elliptic curve $E(\mathbb{Z}/n\mathbb{Z})$ and for a point $P$ computes $Q=kP , k \in \mathbb{N}$ hoping to find the point ...
CommunityBot
- 1
5
votes
1
answer
1k
views
Can the Knaster-Tarski theorem be proved using the Schroeder-Bernstein theorem?
The reverse can be done easily and the proof is well known I am wondering if the exact same argument can be used to prove reverse as well.
CommunityBot
- 1
2
votes
1
answer
583
views
What are the origin and applications of this result?
In a course taught by Morris Eaton on multivariate statistics that dealt mostly with the Wishart distribution, I learned this proposition: Suppose
$$ M = \begin{bmatrix} A & B \\\\ B^T & C \...
CommunityBot
- 1
15
votes
3
answers
1k
views
How to distinguish division algebras from matrix algebras?
Suppose $D$ is an explicitly given rank 9 central simple algebra over ${\mathbb Q}$ (or a number field). For example it could be specified by two cubic subfields ${\mathbb Q}(a), {\mathbb Q}(b)\subset ...
- 1,661
5
votes
0
answers
3k
views
Matrix normalization for constant row and column sums
From a mxn matrix M with only strictly positive elements, I want to multiply each row and each column by real values in order to obtain a matrix N whose row sums are all 1/m and column sums are all 1/...
- 51
3
votes
0
answers
515
views
What happens geometrically when you take associated-graded (or complete, ...) of a group ring at its augmentation ideal?
I am interested in the following functor from Monoids (in $\text{Set}$) to Graded Lie Algebras (over a fixed field of characteristic $0$). (By "graded" I mean only that my Lie algebras have some ...
CommunityBot
- 1
2
votes
0
answers
142
views
Formal solutions of semiring equations
I am looking for a general theorem which would tell me when a formal series solution exists for an equation over a semiring. One may assume that the semiring is equipped with a (formal) derivative. ...
- 11.8k
0
votes
1
answer
287
views
algebraic closure of a subgroup of GL
Let $\Gamma$ be a nontrivial subgroup of $GL(n,\mathbb{C})$. Then does the linear span of $\Gamma$ over $C$ have to contain the algebraic closure of $\Gamma$? Or is there some required condition in ...
- 431
5
votes
0
answers
445
views
symplectic matrices
If $( A B | C D)$ is a symplectic matrix with entries in the finite field with two elements, is it necessarily the case that
$\sum_{i,j,k} a_{ij}b_{ij}c_{ik}d_{ik} = 0$?
This arose in connection ...
- 2,018
3
votes
2
answers
379
views
Partitioning a matrix with bounded row sums
Let $A$ be a $n \times n$ matrix with non-negative entries $a_{ij}$, where $a_{ij}$ is the entry in the $i^{th}$ row and $j^{th}$ column. Assume $\sum_{1 \leq j \leq n} a_{ij} \leq 1$ for all $1 \leq ...
- 501
5
votes
1
answer
1k
views
Orderings of ultrafilters
Let $U$ is a set. I will speak about filters on this set.
If $f$ is a function and $a$ is a filter then I define $f \left[ a \right]$ as
the filter whose base is $\lbrace f[A] | A \in a \rbrace$.
I ...
- 236k
4
votes
2
answers
483
views
Rank of sum of Galois conjugates of a matrix
Given an invertible square matrix $M$ with entries from some number field $K$ which is Galois over $\mathbb{Q}$, sum the Galois conjugates of $M$ to form a new matrix $M' = \Sigma_{\sigma \in \mathrm{...
- 1,810
5
votes
1
answer
560
views
What is the Schouten bracket for the Chevalley-Eilenberg complex with coefficients in a nontrivial module?
Let $\mathfrak g$ be a Lie algebra. The Chevalley-Eilenberg complex is defined to be $\wedge^* \mathfrak g$ with differential $d\colon \wedge^* \mathfrak g\to \wedge^{*-1}\mathfrak g$ defined by $$d(...
- 4,898
2
votes
3
answers
285
views
is there any efficient way to compute the follow matrix equations easily
Let $A$ and $D$ are $n\times n$ diagnal matrices, and $B$ is an $n\times n$ orthogonal matrix. Is there any efficient way to compute the follow matrix equations easily?
$\sum_{i=0}^{k} A^i \cdot B^T \...
- 391
25
votes
8
answers
6k
views
What is the "right" definition of a ring?
This is somewhat related to Greg's question about groups and abelian groups. Suppose you met someone who was well-acquainted with groups, but who was unwilling to accept rings as a meaningful object ...
CommunityBot
- 1
12
votes
2
answers
2k
views
Topological Rings
Is it true that, if S is a subring of a separable topological Noetherian ring R,
then S is separable, too ?
- 1,615
2
votes
0
answers
148
views
Is the homotopy of a primitively generated Hopf algebra still primitively generated?
Let $A=\oplus A_n$ be a primitively generated graded Hopf algebra, where each $A_n$ is a simplicial group. This allows us to define the homotopy group $\pi_*(A)$.
Question: is the graded Hopf algebra ...
- 681
18
votes
1
answer
1k
views
Commuting unitaries
Is the following true:
For every unit vectors $x_1,..., x_n$, $y_1,..., y_n$ in $\mathbb{C}^k$
there exist a Hilbert space $H$, unitary operators $U_1,...,U_n$ and $V_1,...,V_n$ in $B(H)$ and unit ...
- 4,705
3
votes
0
answers
307
views
Construction of an algebra with prescribed representation of the automorphism group.
For this discussion, $G$ is a compact semisimple Lie Group.
For many of the common representations of compact groups, there is a realization of the representation as the automorphisms of some ...
- 5,191
-1
votes
1
answer
185
views
eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I $
Let $A$, $B$ and $C$ be symmetric matrices.
What can we say about eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I $?
- 56.6k
1
vote
0
answers
284
views
Projective but not free (exercise from Adkin - Weintraub) [closed]
This is exercise 38 from Chapter 3. Modules and Vector Spaces in Algebra by Adkins and Weintraub (GTM). How do you solve this problem?
Let
\begin{equation*}
R = \lbrace f : [0, 1] \to \Re : f \;\...
4
votes
1
answer
3k
views
Cauchy-like inequality for Kronecker (tensor) product
General question first: upper/lower bound a sum of Kronecker products by its components. More specifically,
how is $$
\Vert\sum_{\alpha}S_{\alpha}\otimes B_{\alpha}\Vert$$
bounded by the operator ...
- 731
5
votes
1
answer
618
views
Commutator formulas in a universal enveloping algebra
I am interested in finding formulas for commutators of symmetrized monomials in a universal enveloping algebra. Let $C(x_1,\ldots, x_n)= (1/n!)\sum x_{\sigma(1)}\cdots x_{\sigma(n)}$ where the sum ...
- 8,098
1
vote
1
answer
430
views
Another question about primitive central idempotents in associative unital rings (yes, again!)
Let $R$ be an arbitrary associative unital ring and let $x \in R$. Can we always represent $x$ as a finite or infinite sum $x=\sum_{i}y_ie_i$ where $y_i\in R$ and each $e_i$ is a primitive central ...
- 23k
0
votes
1
answer
208
views
I need to prove that any element in a rings is representable as a product of some element and some central idempotent
Let $R$ be an associative ring with identity and let $x$ be an arbitrary element from the ring $R$. Could you please help me to prove that $x=ye$, where $y$ is some element in $R$ and $e$ is some ...
- 47.3k
1
vote
1
answer
600
views
Unimodular column property
Hi, I know that if $R$ is a ring such that every projective $R$-module finitely generated is free then $R$ has the unimodular column property.
I would like to know if there is a ring $R$ that doesn't ...
- 23k
4
votes
1
answer
1k
views
dominant eigenvector
Hi, everyone! Is there any efficient way to simplify the following tensor product
$X \otimes X + X^T \otimes X^T$, where $X$ is a square $n \times n$ matrix.
My goal is to efficiently compute the ...
- 20.2k
1
vote
2
answers
1k
views
trace zero elements in algebraic number fields
If we are given an algebraic number field L, and $ \alpha $ is an element of L whose field trace over Q is zero and whose field norm over Q belongs to Z, then does $ \alpha $ necessarily belong to the ...
- 263
5
votes
2
answers
370
views
M_2(k) as a central extension
Does there exist a field $k$ and a subring $R$ of $S = M_2(k)$ such that $R$ is not finitely generated over its center, $S=kR$ and $1_R = 1_S$? ($S$ is the algebra of $2 \times 2$ matrices over $k$.)
- 361
13
votes
4
answers
3k
views
What is a "block" in an abelian category?
In the literature and in some posts here, there has been variation in the undefined use of the term "block" for a category of modules over a ring, or more abstractly an abelian category (all of which ...
- 1,335
2
votes
0
answers
240
views
Radon transform and Log-concavity
This question is related to (but different from) that of Darsh Ranjan.
Is there a characterization of the functions $f:\mathbb R^n\rightarrow\mathbb R_{\ge0}$ whose Radon transform $\hat f(\omega,t)$...
CommunityBot
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0
votes
1
answer
1k
views
Whether the system of matrix equations is always solvable
In recent days, I learned a linear algebra problem from one of my friends.
It can be stated as follows.
Given four matrices $A,B,C,D$, find three matrices $E,G,F$, not simultaneously zero, such that ...
- 4,705
1
vote
0
answers
334
views
Signature of quadratic form associated to an integral circulant matrix with only real eigenvalues
I am stil stuck with the following:
Let $C$ be a symmetric circulant matrix with integer coefficients of order $n=4k$
(e.g., $C=circ(-1,1,1,1)).$
Assume that $C^{-1}$ is a polynomial (with ...
- 1,641
2
votes
3
answers
12k
views
How to compute the induced $||\cdot||_{2} $ matrix norm of an SPD matrix
Hi, I know they are related questions on the board but mine is more specific. Although the answer for any non-singular matrix would be also interesting. Thanks!
UPDATE: I am sorry I though this ...
- 28.6k
2
votes
2
answers
389
views
Related to fractional ideals
$K$ a field, $A\subset K$ a subring, $M\subset K$ an $A$-submodule. Define
$$(A:_{K}M):= \lbrace s\in K|sM\subset A\rbrace$$
Then it is easy to see that
$$M\subset A\Longleftrightarrow A\subset (A:_{...
- 11
6
votes
2
answers
1k
views
Commutative Ring of Finite Global Dimension
The only examples of commutative rings of finite global dimension I know are either:
Dedekind domains (and fields as a degenerate special case)
Regular local rings
Rings constructed from the previous ...
- 35.2k
18
votes
1
answer
1k
views
Analogue of Smith normal form for matrices over $\mathbb Z[t]$
Let $R$ be a principal ideal domain and $A \in M_n R$. It is well known that there exist invertible matrices $Q$ and $S$ and a diagonal matrix $D= {\rm diag}(a_1,\dots,a_n)$ such that
$a_i \mid a_{i+...
- 545
3
votes
2
answers
2k
views
Memory efficient matrix multiplication
What is the most memory efficient algorithm for calculating $A\cdot B$, where $A,B\in \mathbb{R}^{n \times n}$?
The result of this multiplication might be stored in one of the given matrices ($A$ or $...
- 96.4k
2
votes
1
answer
1k
views
monoid ring and some structure within it - how is it called?
I am amateur - mathematics is my hobby, and I find some strange structure working with toy matrices structure so I try to ask some questions regarding it. Let me allow to introduce some structure ...
- 6,870
30
votes
4
answers
4k
views
Adjacency matrices of graphs
Motivated by the apparent lack of possible classification of integer matrices up to conjugation (see here) and by a question about possible complete graph invariants (see here), let me ask the ...
CommunityBot
- 1
2
votes
1
answer
3k
views
Is it possible to decompose a symmetric, positive definite matrix in this way?
Let $\Sigma$ be a symmetric positive definite matrix. Then the Cholesky decomposition gives us $\Sigma=LL'$ where $L$ is lower triangular and unique.
Under what conditions (if any) does there exist ...
- 96.4k
3
votes
1
answer
383
views
Matrices as dynamical systems
Matrices can be understood in different ways, e.g.
Linear systems of equations
(rich algebraic structure of) Linear mappings
Graphs
Evolution law of discrete-time Dynamical system
Well, 1. und 2. ...
- 39k
9
votes
2
answers
622
views
Is there any transitivity for separable algebras?
If $R$ is a commutative ring (with $1$), then an $R$-algebra $A$ is said to be separable if $A$ is projective as an $A$-$A$-bimodule. (The notion of an "$A$-$A$-bimodule" includes the requirement that ...
- 47.3k
2
votes
4
answers
2k
views
Semi-simple matrices over fields of finite characteristic
Well-known and useful facts are:
any symmetric matrix over $\mathbb R$ is semi-simple (i.e. diagonalizable), and
any hermitean matrix over $\mathbb C$ is semi-simple.
I will loosely speak about the ...
- 9,389
1
vote
2
answers
364
views
Rig of fractions, including zero denominators
For some integral domain $R$, one forms the field of fractions $R^*$ by considering (equivalence classes of) formal pairs {$r/s : r \in R, s\in R\backslash 0$} and defining $+$ and $*$ as you'd expect ...
- 18.7k
1
vote
0
answers
500
views
Dieudonné and generators of the orthogonal group
Let $k>0$ be a positive integer and $n=4k.$ A special case of a result of Dieudonné is
that every element $g$ of the orthogonal group in $n$ variables over the rational numbers
$$
G=O(n,\mathbb{Q}...
- 1,641
3
votes
2
answers
529
views
Circulant $\lbrace -1,1 \rbrace $ matrices with eigenvalues seen in first row
For the circulant matrix $C$ of order $n=4$ with first row $[-1,1,1,1]$ say
$$
C = Circ(-1,1,1,1)
$$
we have the equality of vectors
$$
[R(1),R(\omega),R(\omega^2),R(\omega^3)] = c [-1,1,1,1],
$$
...
- 52.3k
0
votes
0
answers
177
views
Exotic isomorphism of matrix rings (2)
Let $R,S,T$ be (associative, with a 1) rings, and $m,n,r,s$ be non-negative integers.
If
(a) $R$ is isomorphic to $M_{m\times m}(T)$ and $S$ is isomorphic to $M_{n\times n}(T)$ and $m\cdot r = n\...
CommunityBot
- 1