All Questions
Tagged with ra.rings-and-algebras linear-algebra
33 questions
33
votes
2
answers
7k
views
Dimension of infinite product of vector spaces
This question is motivated by the question link text, which compares the infinite direct sum and the infinite direct product of a ring.
It is well-known that an infinite dimensional vector space is ...
- 20.8k
62
votes
25
answers
70k
views
Linear Algebra Texts?
Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in very negative ways to ...
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 ...
17
votes
3
answers
905
views
Existence of translation-invariant basis on $C_c(\mathbb R)$
Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...
- 2,071
10
votes
1
answer
923
views
Conjugation between commutative subalgebras of a matrix algebra?
Let $K$ be an algebraically closed field and $M_n(K)$ the $K$-algebra of all matrices $n\times n$ over $K$. If $L$ and $M$ are two isomorphic commutative subalgebras of $M_n(K)$, it is true that there ...
- 545
10
votes
1
answer
807
views
How many Lie and associative algebras over a finite field are there?
This question is related to the following general question:
Given a variety of (non-associative) algebras $\mathcal V$, a finite field $\mathbb{F}_q$, with $q$ elements, and a positive integer $n$, ...
- 398
53
votes
5
answers
5k
views
Does this formula have a rigorous meaning, or is it merely formal?
I hope this problem is not considered too "elementary" for MO. It concerns a formula that I have always found fascinating. For, at first glance, it appears completely "obvious", while on closer ...
- 15.3k
42
votes
4
answers
33k
views
What is the intuition for the trace norm (nuclear norm)?
I will word this question in terms of linear operators acting on $\mathbb{C}^n$ for simplicity. Feel free to provide an answer in terms of more general Hilbert spaces if you think it makes more sense ...
- 539
17
votes
1
answer
2k
views
Hlawka inequality for determinants of positive definite matrices
It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) $...
- 13.4k
14
votes
1
answer
4k
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 ...
- 1,197
12
votes
2
answers
2k
views
Determinant of identity matrix plus Hilbert matrix
I am looking for the determinant
$$ \det(I_n + H_n) $$
where $I_n$ is the $n \times n$ identity matrix and $H_n$ is the $n \times n$ Hilbert matrix, whose entries are given by
$$ [H_n]_{ij} = \frac{...
- 121
10
votes
1
answer
537
views
Coefficient-wise powers of matrices. Reference wanted
Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant i,...
- 6,349
10
votes
1
answer
3k
views
Reverse Minkowski (and related) Determinant Inequalities
For positive semidefinite matrices $A,B,C \in \mathbb{R}^{n\times n}$, the following inequalities are well known:
$$(\det(A+B))^{1/n} \geq (\det A)^{1/n} + (\det B)^{1/n} $$
and
$$\det(A+B+C) + \...
- 716
9
votes
3
answers
2k
views
On similar matrices and polynomial matrices
I'm teaching linear algebra and I'm encountering 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 ...
- 357
8
votes
2
answers
1k
views
Polynomial roots in the ring extension
Let $R$ be a ring with identity (not necessarily commutative) and $R[x]$ be a ring of polynomials over $R$. We say that a ring $S$ is an extension of $R$ if there is a subring $\tilde{R}$ in $S$ ...
8
votes
0
answers
285
views
Matrix decompositions as monoid isomorphisms. Ever considered before?
I've noticed some correspondences between some matrix decompositions and monoid isomorphisms (always to some free commutative monoid), in addition to the one I asked about in a previous question:
...
- 4,943
8
votes
3
answers
825
views
Does a left basis imply a right basis, without AC?
If $_DV_D$ is a $D$-$D$-bimodule, and we have a $D$-basis for $V_D$, do we still need AC to get a $D$-basis for $_DV$?
(The original question appears below. But this shorter question gets at the ...
- 18.7k
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
633
views
Given a rational matrix $Q$, can we generate $\langle Q^{i}(v)\mid i\in\mathbb Z,v\in\mathbb Z^{2}\rangle$ using only non-negative powers of a matrix?
I have copied this question from StackExchange, thank you to those who helped me to improve the question. (apology if you have seen this question already)
Let $Q $ be a matrix in $ \operatorname{GL}(...
- 823
7
votes
1
answer
456
views
Hopfian modules
My question is slightly motivated by basic results in linear algebra, for example, that if $F$ is a field then a surjective linear map $F^n \rightarrow F^n$ is injective. More generally, any ...
- 11.2k
7
votes
2
answers
963
views
Principal ideal ring, does there exist an invertible matrix such that certain matrix is upper triangular?
I asked here on Math Stack Exchange the following question.
Let $R$ be a principal ideal ring. If $A$ is any $p \times q$ matrix over $R$, then does there exist an invertible matrix $U$ in $\text{M}...
- 305
6
votes
2
answers
2k
views
Positive Elements of a $\ast$-Algebra
In a $C^*$-algebra ${\cal A}$, a positive element is a one of the form $aa^*$, for some $a \in {\cal A}$. It is known that the set of positive elements is a cone, and that for $a,b$ two non-zero ...
- 465
6
votes
2
answers
4k
views
Characteristic polynomial of Kronecker/tensor product
This was asked before on stackexchange but no answer was given. The question is the following:
Let $A$ and $B$ be matrices in $GL(n)$ and $GL(m)$ respectively. Their tensor product $A\otimes B$ is ...
- 3,799
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
5
votes
1
answer
194
views
Subalgebras of singular matrices
Is it true that any subalgebra of singular matrices have a common null-vector?
In other words, is it true that, for any subalgebra $\cal S$
of the algebra of linear operators in a finite-dimensional ...
- 3,932
5
votes
3
answers
757
views
Unital *-homomorphisms between matrices
It is mentioned on Wikipedia that every unital *-homomorphism $\Phi:M_i\to M_j$ is necessarily of the form $\Phi(a)=U^*(a\otimes I_r)U$ for some unitary $U$ and some $r$. (Here $M_i$ are the $i\times ...
- 896
4
votes
1
answer
202
views
Subalgebras of singular matrices (less naive version)
Is it true that, for any subalgebra $\cal S$
of the algebra of linear operators in a finite-dimensional vector space $V$ over a field,
$$
\bigcap_{A\in\cal S}\ker A=\{0\}\hbox{ and }
\bigcup_{A\in\cal ...
- 3,932
4
votes
1
answer
153
views
Linear maps preserved by algebra automorphisms
Let $A$ be a finite dimensional , associative, unital $F$-algebra, where $F$ is a field.
Let $s_A:A\to F$ be an $F$-linear map.
Now consider an arbitrary field extension $K/F$, and define $s_{A\...
- 1,766
3
votes
3
answers
357
views
Can a block matrix with at least 3 zero blocks of different size on the diagonal and 1's everywhere else have only integer eigenvalues?
Let $M=\begin{pmatrix}
\begin{array}{cccccccc}
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
1 & 1 & 0 & 0 & ...
2
votes
1
answer
290
views
Endomorphism rings of infinitely generated free modules generated by idempotents?
Let $M$ be a free right $R$-module. When $M_R\cong R_R^n$ with $n\in \mathbb{Z}_{\geq 1}$, then we know that the endomorphism ring $E={\rm End}(M_R)$ is isomorphic to $\mathbb{M}_n(R)$. We also know ...
- 23
0
votes
0
answers
643
views
A new generalization of the dimension?
During my research, I came a cross on these notions :
Definition 1: A structure $S$, is a pair $(X, \mathcal T)$ with $X$ a set and $\mathcal T$ a set of subsets of $X$, stable by arbitrary ...
- 4,074
0
votes
1
answer
328
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 $K$ ?
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