All Questions
Tagged with ac.commutative-algebra linear-algebra
190 questions
53
votes
9
answers
13k
views
Is there a preferable convention for defining the wedge product?
There are different conventions for defininig the wedge product $\wedge$.
In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$,
in Spivak, we find $\alpha\wedge\beta:=\frac{(k+l)...
- 4,306
49
votes
14
answers
21k
views
Applications of the Cayley-Hamilton theorem
The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own ...
38
votes
2
answers
6k
views
Over which fields are symmetric matrices diagonalizable ?
The question is motivated by this one real symmetric matrix has real eigenvalues - elementary proof:
Are there other fields $F$ than $\mathbb{R}$ (maybe some valued fields or real closed fields) ...
- 567
38
votes
1
answer
10k
views
Infinite tensor products
Let $A$ be a commutative ring and $M_i, i \in I$ be a infinite family of $A$-modules. Define their tensor product $\bigotimes_{i \in I} M_i$ to be a representing object of the functor of multilinear ...
- 63.1k
36
votes
3
answers
2k
views
Are large powers of polynomials linearly independent?
Let $P_1,\dots,P_k$ be polynomials over $\mathbf{C}$, no two of them being proportional.
Does there exist an integer $N$ such that $P_1^N,\dots,P_k^N$ are linearly independent?
- 4,653
34
votes
8
answers
4k
views
Uncountable counterexamples in algebra
In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a ...
30
votes
2
answers
2k
views
When is $SL(n,R) \rightarrow SL(n,R/q)$ surjective?
Let $R$ be a commutative ring with unit and let $q$ be an ideal of $R$. There is thus a natural map $SL(n,R) \rightarrow SL(n,R/q)$ for all $n$. This map is surjective if $SL(n,R/q)$ is generated by ...
- 418
28
votes
5
answers
4k
views
Does Smith normal form imply PID?
Let $R$ be a nonzero commutative ring with $1$, such that all finite matrices over $R$ have a Smith normal form. Does it follow that $R$ is a principal ideal domain?
If this fails, suppose we ...
user5810
28
votes
6
answers
5k
views
Expressing $-\operatorname{adj}(A)$ as a polynomial in $A$?
Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\operatorname{det}(A-xI) = p_0 + p_1x + \dots + p_n x^n$.
I ...
- 427
22
votes
3
answers
2k
views
Discriminant of characteristic polynomial as sum of squares
The characteristic polynomial of a real symmetric $n\times n$ matrix $H$ has $n$ real roots, counted with multiplicity.
Therefore the discriminant $D(H)$ of this polynomial is zero or positive.
It is ...
- 8,086
20
votes
2
answers
1k
views
a determinantal identity
Dusan Pokorny and Jan Rataj have just posted a paper (http://arxiv.org/abs/1209.2305) in which they prove the identity
$$
\det (A-B) = \frac 1{d!} \sum_{k=0}^d (-1)^k \binom dk \det((d-k)A + kB)
$$
...
- 340
19
votes
1
answer
4k
views
How should I think about the module of coinvariants of a $G$-module?
Let $G$ be a group, $M$ a $G$-module, then the group of coinvariants is the module $M_G := M/I_GM$, where $I_G$ is the kernel of the augmentation map $\epsilon : \mathbb{Z}G\rightarrow \mathbb{Z}$.
...
- 7,527
18
votes
2
answers
1k
views
Is a matrix similar to its transpose over $\mathbb{Z}_p$?
Is every $n \times n$ matrix with entries in $\mathbb{Z}_p$ (or even $\mathbb{Z}$) conjugate to its transpose via a matrix in $GL_n(\mathbb{Z}_p)$?
On the one hand, I know the analogous fact is false ...
- 2,242
16
votes
3
answers
2k
views
Hom(A,C) ⊗ Hom(B,D) injects into Hom(A⊗B,C⊗D): when? why?
Sorry for asking a linear algebra question on a research forum, but this seems to be either a case of extreme blindness on my side, or a case of a result lying much deeper than it seems.
The ...
- 33.8k
16
votes
1
answer
711
views
A weird question about two weird decompositions of $\mathbb{R}$ as a $\mathbb{Q}$-vector space
While working in a question about the affine group $\text{Aff}(\mathbb{R})$, I have come up with the following strange question about the real numbers:
Question: Do there exist a non-trivial ...
- 1,101
16
votes
1
answer
2k
views
Commuting Matrices and the Weak Nullstellensatz
In the Wikipedia article on Hilbert's Nullstensatz,
http://en.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz
the following application of the Weak Nullstensatz is mentioned:
Commuting matrices
...
- 839
15
votes
1
answer
679
views
Submodules of $({\mathbb Z}/6{\mathbb Z})^n$ intersecting $\{0,1\}^n$ trivially
$\newcommand{\F}{{\mathbb F}}$
$\newcommand{\Z}{{\mathbb Z}}$
Suppose that $\F$ is a finite field of prime order $p:=|\F|$, and let $n$ be a positive integer. I consider the regime where $\F$ is ...
- 23k
14
votes
4
answers
6k
views
When is an algebra of commuting matrices (contained in one) generated by a single matrix?
Let C be an nxn matrix, then the polynomials in C (with appropriate coefficients) form an algebra of commuting matrices. I feel that I should know if the converse is true but I do not. So my first ...
- 30.1k
14
votes
1
answer
445
views
Similar matrices over $\mathbb Z_p$
Let $A$ and $B$ be two $n \times n$ matrices with entries in $\mathbb Z_p$, the $p$-adic integers. Is it true that $A$ and $B$ are conjugate iff they're conjugate over $\mathbb Q_p$ and over $\mathbb ...
- 683
14
votes
1
answer
2k
views
Why does this matrix have zero determinant?
This curious identity arose from studying reductions of the maximal ideal in certain monomial algebra. It can be proved "by hand", (i.e, using Macaulay 2), but I am seeking a more conceptual ...
- 30.5k
14
votes
0
answers
602
views
Is the Zariski density proof of Cayley-Hamilton circular?
This old MO thread and its comments contains a discussion of the Zariski density proof of Cayley-Hamilton (I have also asked a separate question about the proof Victor gives in the comments here). ...
- 118k
13
votes
2
answers
1k
views
Is the Characteristic of a Field Detectable from the Topology of a Topological Vector Space?
Motivation
A topological vector space is a vector space over a (topological) field, K, that carries a topology such that addition and scalar multiplication are continuous maps, e.g., all normed vector ...
13
votes
1
answer
496
views
Functorial multiplication on commutative rings
Let $C$ be the category of associative commutative rings with 1 and let $F:C\to C$ be a functor which commutes with the forgetful functor to abelian groups (i.e. $F$ is a functorial way to define ...
- 7,037
12
votes
4
answers
752
views
Additive commutators and trace over a PID
I would like to find an example of principal ideal domain $R$, such that there exists a square matrix $A\in \mathfrak{M}_n(R)$ with zero trace that is not a commutator (i.e. for all $B,C \in \mathfrak{...
- 2,829
10
votes
4
answers
2k
views
Formal power series is Taylor expansion of rational function iff Hankel determinants vanish?
Let $$ u(T)=\sum_{n = 0}^\infty a_nT^n$$ be a formal power series over a field $K$. Then why does $u(T)$ lie in $K(T)$ (i.e. is the Taylor expansion of a rational function) if and only if there is an $...
user97565
10
votes
2
answers
2k
views
Characteristic polynomial of exterior power
Suppose $f$ is a linear map, and consider $\Lambda^k f$ as the usual exterior power of $f$ (if you prefer matrices, it is a matrix whose entries are the $k\times k$ minors of $f.$) The coefficients of ...
- 96.4k
10
votes
0
answers
312
views
Triangle $X'\to X\to X''\to\Sigma X'$ splits if $X\simeq X'\oplus X''$?
Given a commutative ring $R$ and a distinguished triangle $X'\to X\to X''\xrightarrow e\Sigma X'$ in the derived category $D(R)$, where $X',X,X''$ are perfect complexes. If we have an equivalence $X\...
user20948
9
votes
2
answers
2k
views
Jordan Form Over a Polynomial Ring
Let $X$ be the set of $k\times k$ matrix with entries in $\mathbb{C}$, and let $M\in X$. The group $GL(k,\mathbb{C})$ acts on $X$ by conjugation, and according to the Jordan decomposition theorem (...
- 2,087
9
votes
2
answers
900
views
Compute adjugate matrix over commutative ring
Let $A$ be a $n\times n$ matrix over a commutative ring. I'm looking for a good method to compute its adjugate matrix.
My current approach is to use the Cayley-Hamilton theorem:
$$\text{adj}(A) = -(A^...
9
votes
3
answers
1k
views
Vector spaces with natural bases
Sergeib's question asks about vector spaces without a natural basis.
Actually, I would claim (apparently in accord with many comments and answers to Sergeib's question ) that this is the default ...
9
votes
1
answer
296
views
What are the points of the algebra of polynomial functions on an arbitrary vector space?
Let $V$ be an arbitrary vector space over some field $\mathbb{K}$ (UPD: of characteristic 0), $V^*=\mathrm{Hom}(V,\mathbb{K})$ its linear dual. Let $\mathrm{Sym}_\mathbb{K}(V^*)$ be the free ...
- 480
8
votes
1
answer
604
views
Number of zeros of the derivatives of a polynomial
What is the maximum total number of zeroes a univariate polynomial $f\in\mathbb{C}[z]$ of degree $d$, together with all of its derivatives, can have at $k$ given points of $\mathbb{C}$?
I am ...
- 2,179
8
votes
3
answers
740
views
Is there some example that nicely extends the multiplication of natural numbers?
Motivation: In mathematics, it is natural to decompose a complicated thing into simpler ones. In the system of natural numbers, the process to decompose large number is to factor it. The ...
- 181
8
votes
1
answer
641
views
Property of the trace on finitely generated projective modules
Let $A$ be a commutative ring with unit and let $P$ be a projective $A$-module finitely generated. By definition, there exists an $A$-module $P'$ such that $P\oplus P'$ is free of finite rank $r$. If $...
- 1,479
8
votes
2
answers
406
views
Matrix diagonalization and eigenvector computation constructively
Assuming Bishop's constructive mathematics, is it true that any real-valued square matrix with distinct roots of the characteristic polynomial can be diagonalized? By distinct, I mean apart: $x \neq y ...
- 791
8
votes
2
answers
425
views
Dimension of commutative subalgebras of a central simple algebra
let $k$ be a field, and let $A$ be a central simple $k$-algebra over $k$.
What is the maximal dimension of a commutative $k$-subalgebra of $A$?
If $A=M_r(D)$, where $D$ is a central division $k$-...
- 81
8
votes
1
answer
506
views
The anti-symmetrization of a kind of polynomials in $\mathbb{Z}[x_1,x_2,\ldots,x_n]$
Let $n$ be a positive integer and $S_n$ be the symmetric group on $\{1,2,\ldots,n\}$. Let $\mathcal{A}_n$ be the anti-symmetrization operator on $\mathbb{Z}[x_1,x_2,\ldots,x_n]$ such that for any $f(...
- 1,997
8
votes
1
answer
353
views
$E_n(\ell^\infty)=SL_n(\ell^\infty)$?
Let $R$ be a commutative unital ring $R$ with unit element $1$.
For $n\in \mathbb{N}=\{1,2,3,\cdots\}$, let $SL_n(R)$ be the group of all $n\times n$ matrices with entries from $R$ having ...
- 81
8
votes
1
answer
202
views
Is there a ring which is not Hermite but is coherent?
Call a commutative unital ring $R$
Hermite if for all $m, n\in \mathbb{N}$ with $m<n$, and all $f\in R^{m\times n}$ such that transpose($f$) is left invertible (with a matrix with entries from $R$ ...
- 85
8
votes
0
answers
292
views
Image of multiplication map in tensor powers of finite-dimensional ring
Let $R$ be a (commutative, unital) ring of dimension $n$ over a field $k$. Assume the characteristic of $k$ is greater than $n$.
Then $R^{\otimes n}$ has a natural ring structure, together with an $...
- 148k
7
votes
2
answers
942
views
Hilbert's Nullstellensatz on polynomials with integer coefficients
Let $f_1, f_2, \ldots, f_m \in \mathbb{Z}[x_1, \ldots, x_n]$. Assume $f_1(X) = f_2(X) = \ldots = f_m(X) = 0$ have no solutions over $\mathbb{C}^n$, then by Hilbert's Nullstellensatz, there exists ...
- 651
7
votes
2
answers
311
views
The coefficient of a specific monomial in the expansion of the following polynomial
Let $a_{n,k}$ be the coefficient of $$X_1^{\frac{k(n-1)}{2}}X_2^{\frac{k(n-1)}{2}}\cdots X_n^{\frac{k(n-1)}{2}}$$ in the expansion of the real polynomial $$\left(\prod\limits_{1\leq i<j\leq n}(X_j-...
- 1,997
7
votes
3
answers
2k
views
Is there a field which is the union of finitely many proper subfields?
Is there a field which is the union of finitely many proper subfields?
- 79
7
votes
1
answer
3k
views
When does the determinant distribute over addition?
When does $\det(A+B)=\det(A)+\det(B)$ hold?
I actually wonder if there is an easy answer for when $Per(A+B)=Per(A)+Per(B)$.
- 13.9k
7
votes
0
answers
225
views
Decomposing an endomorphism as a tensor product
$\DeclareMathOperator\End{End}$Let $f$ be an endomorphism of the finite-dimensional vector space $V$, over the field $K$. The question of whether $f$ is decomposable, that is, whether $V$ can be ...
- 2,287
6
votes
2
answers
5k
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 $\mathbb{...
- 2,829
6
votes
1
answer
328
views
Algebra with a certain abelian group as the multiplicative group
Let $A$ be an abelian group. Are there an algebra $\mathfrak{X}(A)$ s.t. the multiplication group is isomorphic to A ? i.e.
$$
\mathfrak{X}(A)^{\times} \simeq A.
$$
For example, for $A=\mathbb{Z}/4\...
- 479
6
votes
1
answer
725
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 ...
- 7,500
6
votes
3
answers
1k
views
Complexity of solving systems of linear diophantine equations
It is "well known" that a matrix system $Ax=b$ where $A\in \Bbb Z^{m\times n}$, $x\in \Bbb Z^n,b\in\Bbb Z^m$ for some $m,n \in \Bbb N$, can be solved in polynomial time, using Smith/Hermite Normal ...
- 972
6
votes
1
answer
726
views
Jordan form on an invariant vector subspace
Let $\mathbb{F}$ be a field and $V$ an $\mathbb{F}$-vector space. Let $\operatorname{T}\in\mathrm{End}(V)$ be an $\mathbb{F}$-linear operator. It is well known that if $\dim V<\infty$ then $\...
- 1,959