# Tagged Questions

**2**

votes

**0**answers

69 views

### Separating duality for TVS?

What is the modern concept (term) for "separating duality" (dualité séparante in french) in the sense of Bourbaki (TVS Ch II § 6) as explained in the following ?
...

**2**

votes

**1**answer

322 views

### Who defined and who coined “module”?

The title of my Q. says it all:
QUESTION: Who defined and who coined: module?
Would it be Emmy Noether?
EDIT In view of @anon's and KConrad's answers, and as it could have been ...

**3**

votes

**1**answer

122 views

### What is a degenerate Legendre Transformation?

I am studying the Lagrangian and Hamiltonian description of some dynamical systems. The problem with this description of the particular kind of systems I am studying, is that the Legendre ...

**4**

votes

**1**answer

150 views

### Isomorphism of matrix ring over ore domain

Let $R_1,R_2$ be (left and right) ore domains. Does $ Mat_n(R_1)\cong Mat_m(R_2)$ implie m=n and $q.f.(R_1)\cong q.f.(R_2)$?
An counter example, a proof or a reference is welcomed.
Thanks

**2**

votes

**2**answers

344 views

### Central idempotents from characters in Frobenius algebras (generalizing Lusztig arXiv:math/0208154v1 §19)

$\newcommand{\refone}{\textbf{(1)}}$ $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\tr}{\operatorname{Tr}}$ $\newcommand{\kk}{\mathbf{k}}$ Let $\kk$ be a field. Let $A$ be a $\kk$-algebra which ...

**15**

votes

**5**answers

2k views

### Origin of exact sequences

I have seen exact sequences appearing a lot in algebraic texts with different purposes. But I've never seen names of the people associated with it. Also I don't understand what's so good about showing ...

**2**

votes

**1**answer

70 views

### Rings in which every J-matrix is non-singular

Let $R$ be a ring with identity. A matrix $A=[a_{ij}] \in M_n(R)$ is called a J-matrix if for any $i$, $a_{ii} \not \in J(R)$ but for any $i \not = j$, $a_{ij} \in J(R)$. Now suppose that every ...

**0**

votes

**1**answer

109 views

### On separable field extensions [closed]

Let $F\subseteq K$ be a finite separable field extension with $a_1,..., a_n$ an $F$-basis for $K$. Is it true that the matrix $A := [\mbox{tr}(a_ia_j)]$ is non-singular ?

**4**

votes

**0**answers

221 views

### Hermitian forms over quaternion algebra

Notations: Let $Q=(a,b)$ be a quaternion algebra over a field of characteristic $\neq 2$, i.e. $i^2=a, j^2=b, k=ij, ij=-ji$. Consider $K=k(t)(\alpha)$, where $\alpha=\sqrt{at^2+b}$. Let ...

**11**

votes

**2**answers

1k views

### Writing a matrix as a sum of two invertible matrices

Let $n\geq 2$. Is it true that any $n\times n$ matrix with entries from a given ring (with identity) can be written as a sum of two invertible matrices with entries from the same ring ?

**0**

votes

**1**answer

170 views

### Number of Minimal left ideals in the full matrix ring over a finite commutative local ring

Inspired with another QUESTION I would like to know the number of minimal left ideals of $M_n(R)$ in terms of $n$ and $R$ where $R$ is a finite local commutative ring with identity ?

**0**

votes

**1**answer

114 views

### Number of minimal left ideals

Is there any way to compute the number of minimal left ideals of $M_n(K)$, the full $n\times n$ matrix ring with entries in the field $K$ ?

**4**

votes

**0**answers

105 views

### What is the composition in SesquiAlg?

To motivate my question, I will describe a famous 2-category. First, there is the 1-category $\text{Vect}$ of vector spaces (over some fixed field). This category has a tensor product $\otimes$, and ...

**3**

votes

**1**answer

186 views

### Existence of a generalized matrix inverse over an arbitrary field?

Let $A\in M_n(K)$ be a square matrix over a field $K$. The notion of inverse matrix was generalized by Moore and Penrose for real and complex matrices
(also called pseudo-inverse $A^{\dagger}$ of $A$, ...

**5**

votes

**0**answers

198 views

### On the linear transformation between matrix space

Suppose we have two matrix subspaces, $n\times n$ matrix subspace $S_1$ and $m\times m$ matrix subspace $S_2$. Every element of $S_1$ and $S_2$ is complex symmetric matrix.
Suppose there exists ...

**12**

votes

**0**answers

515 views

### How to explain the picturesque patterns in François Brunault's matrix?

How to explain the patterns in the matrix defined in François Brunault's
answer to the question Freeness of a Z[x] module depicted below? --
Choosing colors according to the highest power of 2 which ...

**2**

votes

**0**answers

112 views

### when upper triangular matrix modulo prime ideals implies upper triangular?

Let $E/ \mathbb{Q}_p$ be a finite extension, let $\mathcal{O}$ be the ring of integers of $E$. Let $A$ be a reduced noetherian local complete $\mathcal{O}$-algebra with the maximal ideal ...

**7**

votes

**1**answer

460 views

### Does this Linear Algebra Construction have a Name?

Let $\mathcal{R}$ be a ring and let $v^0,\ldots,v^{k-1}\in\mathcal{R}^m$ with $m \geq k$. Suppose we wish to find $w\in Span(v^0,\ldots,v^{k-1})$ such that $k-1$ specified coordinates of $w$ vanish ...

**19**

votes

**4**answers

1k views

### The sum of same powers of all matrices modulo p

The following is a problem from our department algebra competition for
students:
Non-question.
An experimental-math geek was trying to raise all matrices $17\times17$
over the field with 17 ...

**0**

votes

**1**answer

155 views

### Tensoring with descending chain of modules

Let $A \to B$ be a ring homomorphism. Let $M_1 \supseteq M_2\supseteq \ldots$ be an infinite chain of $A$-modules ($M_i$ not necessarily finite free). Suppose that the limit $\cap_{i=1}^{\infty} M_i$ ...

**0**

votes

**1**answer

146 views

### Is there a wedge which operates on multiple vector spaces?

Let's say I have two vector spaces $V,W$ , and we have the graded algebras $\Lambda(V),\Lambda(W)$, each with an operation $\wedge$. I'd like to know if there are "many" $\wedge$ operators, or if ...

**1**

vote

**0**answers

96 views

### Complementation in an extension field

If $E$ is an extension field of $F$, is $F$ necessarily (without assuming the axiom of choice) complemented as a vector subspace of $E$? (Of course the answer is easily yes if the extension is ...

**5**

votes

**4**answers

1k views

### Proving a determinant = 0

The two most elementary ways to prove an N x N matrix's determinant = 0 are:
A) Find a row or column that equals the 0 vector.
B) Find a linear combination of rows or columns that equals the 0 ...

**1**

vote

**0**answers

199 views

### Average weighted value of a linear functional over increasing bounded subsets of Z^n

Say you're working within the finite-dimensional free Z-module $\mathbb{Z}^n$, and you want to impose a "norm" on this module. By a "norm" I mean a function $\|·\|: \mathbb{Z}^n \to \mathbb{R}$ which ...

**5**

votes

**0**answers

403 views

### A reference on semisimple linear algebra

Is there any literature where the tools familiar from (multi)linear algebra are systematically transferred to the setting of semisimple modules over noncommutative rings?
In fact this question is a ...

**1**

vote

**1**answer

244 views

### Is the trivial solution the only solution? [closed]

Let n be a positive integer, and c_1, c_2, ... c_n be (unkown) real numbers .
Consider the system
$$c_1+c_2+ ... +c_n=0,$$
$$c_1^2+c_2^2+ ... +c_n^2=0,$$
$$c_1^3+c_2^3+ ... +c_n^3=0,$$
$$.... ...

**7**

votes

**5**answers

1k views

### Centralizer of a Matrix over a Finite Field

This question in stackExchange remained unanswered.
Let $\mathbb F$ be a finite field. Denote by $M_n(\mathbb F)$ the set of matrices of order $n$ over $\mathbb F$ . For a matrix $A∈M_n(\mathbb ...

**7**

votes

**2**answers

881 views

### Urge/reason for inventing interior product ( Grassmann algebra )

Hello everyone,
I wanted to lecture on Grassmann and his works , and I have been reading the collected works of Grassmann " Die Lineale Ausdehnungslehre ". There Grassmann introduced something ...

**3**

votes

**1**answer

354 views

### Relaxing commutativity. For c1,c2 find q1,q2: (1) [c1,c2]=q1c2-q2c1 (2) [q1,q2]=0, (3)…

Consider some elements c1,c2 in some ring.
Let me say that they are "relaxed commutative" if there exists two elements q1,q2,
such that the following conditions hold:
(1) $ [c_1,c_2]=c_1q_2-c_2q_1$
...

**0**

votes

**3**answers

259 views

### Vector “product” diagonalization

Consider a vector space $V$ (with dimension $n+1$ and elements $v$) on which a (commutative and associative) "product" $\odot$ taking $V\odot V\rightarrow V$ is defined, and an $1$ element $v_0$ ...

**2**

votes

**1**answer

167 views

### Solvability of the quaternion equations of the Quaternion rationals

So, Let K be the non-commutative field or division ring of the Quaternions. We will consider a sub-field of it L=Q[i,j] in other words, L=Q+Qi+Qj+Qk. Now let us consider the ring of skew polynomials ...

**4**

votes

**2**answers

827 views

### Vandermonde matrices and general position

I was wondering if it is known whether a Vandermonde matrix over a sufficiently large finite field is in general position with respect to intersections of subspaces spanned by subsets of
columns, i.e. ...

**4**

votes

**1**answer

2k views

### Determinant of a sum of two matrices (one dominating the other)

Let $A$ and $B$ be two $n \times n$ real matrices such that:
$\forall i, j: a_{ij} \geq 0, b_{ij} \geq 0$
let $a_\max$ be the largest entry of $A$ and $b_\min$ be the smallest nonzero entry of $B$; ...

**0**

votes

**1**answer

261 views

### Tensor Products, Sub-Algebras, Sub-Modules, and Inclusions

Let $A$ be a not neccessarily commutative algebra, and let $B \subset A$ be a subalgebra of $A$. Moreover, let $M$ be an $A$-bimodule, and let $N \subset M$ be a $B$-sub-bimodule. The tensor product ...

**2**

votes

**1**answer

167 views

### Indecomposable extensions of regular simple modules by preprojectives

Given four points in general position on $\mathbb{P}^2$ there exists a projection to $\mathbb{P}^1$ collapsing these four pairwise to two points. Its kernel is some fifth point on $\mathbb{P}^2$.
In ...

**1**

vote

**0**answers

127 views

### Diagonalizing matrices of linear forms of indeterminates

Let $B$ be a matrix with elements as linear forms of indeterminates. Is there a proper diagonalization procedure for such matrices like those of matrices with real and complex entries?

**4**

votes

**1**answer

336 views

### Basis of a Finite Dimensional Algebra with a Finitely Generated Relation Set By Computer

Let $A$ be a noncommutative finitely generated algebra with a finitely generated set of relations. Moreover, assume that $A$ is finite dimensional as a vector space.
What I want to know is, can ...

**3**

votes

**3**answers

237 views

### measuring n 2-planes in R^{2n}.

Given $n$ vectors $v_1, \ldots, v_n$ in $\mathbb{R}^n$ of course we all know at least one measure for their relative configuration: $|v_1 \wedge\ldots \wedge v_n|$. Now suppose one were given $n$ ...

**5**

votes

**1**answer

1k views

### Algebra - Decomposition of a matrix polynomial

Dear All,
This is related with a problem that I'm trying to solve on my PhD dissertation in econometrics, and I thought that some mathmatician can know the answer.
What is known about a possible ...

**1**

vote

**3**answers

922 views

### Generalization of eigenvalues/vectors to modules?

What is the generalization of eigenvalues/vectors to modules?
To be specific, given a "vector" v in a module over some ring, and a linear "operator" O from the module to itself (please feel free to ...

**0**

votes

**1**answer

226 views

### Chain of ideals in a complex algebra

Suppose $\mathfrak{A}$ is an unital algebra over complex numbers and $\mathfrak{J}$ is chain of left-ideals in $\mathfrak{A}$ ordered by inclusion such that none of its elements is countably ...

**0**

votes

**0**answers

144 views

### Matrices satisfying certain pair-wise constraints

Consider given pairs of variables: $a_{ir1},a_{ir2}\in \mathbb{R}^{m \times m}$ and $a_{jr1},a_{jr2}\in \mathbb{R}^{m \times m}$, where $r \in \{1,2,\cdots,t\}$, consider the constraints:
...

**8**

votes

**1**answer

1k views

### Do these matrix rings have non-zero elements that are neither units nor zero divisors?

First, a disclaimer: This is a repost of a question I asked on stackexchange (no answer there).
Let $R$ be a commutative ring (with $1$) and $R^{n \times n}$ be the ring of $n \times n$ matrices with ...

**3**

votes

**5**answers

424 views

### Is there a name for this map induced by bilinear forms?

Let $V$ be a real vector space. A bilinear form $\langle \rangle:V\times V\to {\mathbb{R}}$ induces a linear functional $\theta$ on the tensor product $V\otimes V$ given by sending the finite sum ...

**5**

votes

**2**answers

423 views

### adjoint of multiplication operator in a commutative algebra

Dan Popescu asked me the following question, and since I'm not an expert I'm throwing his question on MO.
Suppose that $A$ is a finite-dimensional vector space over an ordered field $k$ with ...

**6**

votes

**2**answers

2k views

### Periodic matrices

A square matrix $M$ such that $M^{k+1}=M$, for some positive integer $k$, is called a periodic matrix.
Can we characterize the periodic matrices in $\mathcal{M}_n(\mathbb{Z})$?
If we replace ...

**0**

votes

**2**answers

337 views

### Commutative *-subrings of the noncommutative C*-algebra $B(l^2)$

A $\star$-ring is a ring with an involutive anti-automorphism. The simplest example of a noncommutative $\star$-ring is perhaps $B(l^2)$, the ring of bounded linear functions on the sequence space ...

**2**

votes

**1**answer

283 views

### nth-powers and degree n polynomials with coefficients in field extensions

Hi,
Suppose that $E/F$ is a Galois extension. If $P(X)\in E[X]$ is a (EDIT: monic) polynomial of
degree $n > 0$, such that $P(X)^n\in F[X]$, does it follow that $P(X)\in F[X]$?
Thanks

**6**

votes

**2**answers

2k views

### on similar matrices and polynomial matrices

I m teaching linear algebra and encounter this theorem:
two matrices A and B are similar iff tI - A and tI - B are equivalent (as polynomial matrices), where I is the unit matrix.
The proof that I ...

**17**

votes

**6**answers

10k views

### Determinant of sum of positive definite matrices

Say $A$ and $B$ are symmetric, positive definite matrices.
I've proved that $\det(A+B) \ge \det(A) + \det(B)$ in the case that $A$ and $B$ are two dimensional.
Is this true in general for ...