All Questions
6,260 questions
3
votes
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problems of subspace of M_n(C)
let $M_n(c)$ denote the n times n matrices over the complex number field. $N$ be a subspace of
$M_n(C)$.
1 If there is no unitary lies in $N$, what is the maximum of the dimension of $N$ can be?
...
- 1,503
12
votes
4
answers
1k
views
Topologizing free abelian groups
For any set $S$ one can consider the free abelian group $\mathbb{Z}[S]$ generated by this set. Now suppose, there is a topology on $S$ given. Is it possible to find a topology on $\mathbb{Z}[S]$ in ...
- 11.1k
3
votes
1
answer
1k
views
Matrix approximation
Let A be an $m\times n$ matrix and $k$ be an integer. Assume that $A$ is non-negative. We want to find a scalar $\epsilon$ and an $n\times n$ matrix $B$ such that $A\leq A(\epsilon I + B)$ (where $\...
- 145
1
vote
2
answers
440
views
A problem concerning two symmetric matrices
Let A , B denote two symmetric matrices of the same order n. and Spec(A)=X , Spec(B)=Y.
If Spec(A+B)=X $\cup$ Y , proof thar AB=0.
here Spec(A) means the set of the engevalues of A.
This is a ...
- 363
19
votes
4
answers
2k
views
Problems concerning subspaces of $M_n(\mathbb{C})$
Let $M_n(\mathbb{C})$ denote the n times n matrices over the complex number field. N be a subspace of $M_n(\mathbb{C})$.
If all the matrices in N are non-invertible , what is the maximum the ...
- 363
6
votes
4
answers
17k
views
Linear Regression Coefficients W/ X, Y swapped
Let's say I have a linear regression model of the form $ y = B_x x + I_x + \epsilon $, where $B_x$ is the beta coefficient of the $x$ term, $I_x$ is the intercept term and $\epsilon$ is additive, ...
- 159
4
votes
8
answers
3k
views
Theory of cones
Hi all,
Can anyone point me to some references to the theory of finitely-generated cones in euclidean space? I'd like to know in particular if there is a notion of basis/dimension/linear dependence ...
0
votes
1
answer
8k
views
Product of Positive Matrices
Is the product of non-negative definite matrices also non-negative definite? If not, let A and B be non-negative definite matrices, is '$\operatorname{tr}(A^T B) \ge0$' ?
- 37
12
votes
2
answers
2k
views
Non-degenerate multilinear forms
Is there a standard notion of non-degeneracy for multilinear forms?
My motivation is simple curiosity, by the way!
- 47.3k
7
votes
3
answers
2k
views
Sarrus determinant rule: references, extensions
SEEKING REFERENCES FOR SARRUS' RULE AND EXTENSIONS
An undergraduate came to me with an identity for 4x4 determinants that is actually correct:
$\det(A)=h(A)+h(RA)+h(R^{2}A)$
where R cyclically ...
- 73
12
votes
0
answers
716
views
Lifting abelian varieties in (the closed fiber of) a fixed Neron model
Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth ...
- 1,609
18
votes
3
answers
8k
views
Number of invertible {0,1} real matrices?
This question is inspired from here, where it was asked what possible determinants an $n \times n$ matrix with entries in {0,1} can have over $\mathbb{R}$.
My question is: how many such matrices ...
- 32.1k
30
votes
12
answers
14k
views
Why are tensors a generalization of scalars, vectors, and matrices?
Take two vector spaces $V$ and $W$ over a field $F$. One may form the tensor product $V\otimes W$ and it fulfills an universal property. Elements of $V\otimes W$ are called tensors and they are linear ...
18
votes
3
answers
6k
views
Number of unique determinants for an NxN (0,1)-matrix
I'm interested in bounds for the number of unique determinants of NxN (0,1)-matrices. Obviously some of these matrices will be singular and therefore will trivially have zero determinant. While it ...
- 359
5
votes
3
answers
781
views
Acyclic quivers differing only in arrow directions: functorial isomorphism of representation categories?
Let $Q$ and $R$ be two acyclic quivers which differ only in the directions of their arrows (i. e., the underlying undirected graphs are the same).
1. Does there exist an isomorphism of additive ...
- 33.8k
9
votes
2
answers
3k
views
What tensor product of chain complexes satisfies the usual universal property?
Recall that a chain complex is a (finite) diagram of the form
$$ V = \{ \dots \to V_3 \overset{d_3}\to V_2 \overset{d_2}\to V_1 \overset{d_1}\to V_0 \to 0 \} $$
where the $V_n$ are (finite-dimensional)...
- 54.6k
4
votes
2
answers
402
views
lower bound for torsion of abelian varieties
Let $A$ be an abelian variety defined over a field $K$ of characteristic $p>0$. Let $A[\ell]$ be the group of $\ell$-torsion points, $\ell\neq p$ a prime. Are there positive constants $C, \eta$ ...
38
votes
6
answers
11k
views
Is there a version of inclusion/exclusion for vector spaces?
I am asking for a way to compute the rank of the 'join' of a bunch of subspaces whose pairwise intersections might be non-zero. So in the case n=2 this is just $\dim(A_1+A_2) = \dim(A_1) + \dim(A_2) - ...
- 549
4
votes
0
answers
97
views
bounded homogeneous quartics
If Q is a real homogeneous quartic on $R^N$,
$Q(x) = \sum_{1 <= i,j,k,l <= N} Q_{ijkl} x_i x_j x_k x_l$
what is the condition on the (totally symmetric) coefficients $Q_{ijkl}$ for Q ...
- 362
4
votes
2
answers
236
views
Order of "one minus automorphism"
This is something I am stuck on (it might well be trivial- in which case this is an embarassing question):
Let V be a dimension r vector space over Fp, the field with p prime elements (I also care ...
- 22.2k
66
votes
3
answers
4k
views
Does linearization of categories reflect isomorphism?
Given a category $C$ and a commutative ring $R$, denote by $RC$ the $R$-linearization: this is the category enriched over $R$-modules which has the same objects as $C$, but the morphism module between ...
- 6,162
10
votes
4
answers
8k
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Any reference on multilinear algebra [closed]
Do you know any good reference on multilinear algebra?
3
votes
1
answer
589
views
A question on star-congruence.
We consider $n\times n$ complex matrices. Let $i_+(A), i_-(A), i_0(A)$ be the number of eigenvalues of $A$ with positive real part, negative real part and pure imaginary. It is well known if two ...
- 1,858
2
votes
1
answer
2k
views
Geometric interpretation of singular values
The singular values of a matrix A can be viewed as describing the geometry of AB, where AB is the image of the euclidean ball under the linear transformation A. In particular, AB is an elipsoid, and ...
- 107
21
votes
5
answers
2k
views
The middle eigenvalues of an undirected graph
Let $ \lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_{2n} $
be the collection of eigenvalues of an adjacency matrix of an undirected graph $G$ on $2n$ vertices. I am looking for any work or references ...
- 784
3
votes
2
answers
733
views
If the morphism of root data induced by an isogeny of a reductive group is a Frobenius, is then the isogeny itself a Frobenius?
Let $G$ be a reductive (or just semisimple) algebraic group over an algebraically closed field $k$ of characteristic $p > 0$, let $T$ be a maximal Torus and let $f:G \rightarrow G$ be an isogeny. ...
- 5,243
2
votes
2
answers
536
views
Sequence of constant rank matrices
Let $A_k$ be a sequence of real, rank $r$, $n$ x $m$ matrices such that $A_k$ converges to a rank $r$ matrix $A$. Let $v_k, u_k$ be sequences of vectors such that $u_k\rightarrow u$ and $A_k v_k=u_k$. ...
- 1,638
11
votes
2
answers
5k
views
Quadratic forms over finite fields
I'm reading some very old papers (by Birch et al) on quadratic forms and I don't get the following point:
If $f$ is a quadratic form in $X_1,X_2,\cdots,X_n$ over a
finite field, then one can ...
- 5,163
57
votes
21
answers
14k
views
Linear algebra proofs in combinatorics?
Simple linear algebra methods are a surprisingly powerful tool to prove combinatorial results. Some examples of combinatorial theorems with linear algebra proofs are the (weak) perfect graph theorem, ...
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 ...
7
votes
2
answers
1k
views
Hochschild Kostant Rosenberg theorem for varieties in positive characteristic?
Is there is a known version of the HKR theorem as proved in say Swan's paper "Hochschild Cohomology of Quasiprojective Varieties" in positive characteristic? I assume something is known about this as ...
- 3,758
81
votes
10
answers
9k
views
Existence of a zero-sum subset
Some time ago I heard this question and tried playing around with it. I've never succeeded to making actual progress. Here it goes:
Given a finite (nonempty) set of real numbers, $S=\{a_1,a_2,\dots, ...
- 85.6k
22
votes
2
answers
4k
views
Fast Fourier transform for graph Laplacian?
In the case of a regularly-sampled scalar-valued signal $f$ on the real line, we can construct a discrete linear operator $A$ such that $A(f)$ approximates $\partial^2 f / \partial x^2$. One way to ...
- 3,059
3
votes
1
answer
538
views
Non-negative matrices with prescribed Perron-Frobenius eigenvectors
In my research I came across the following question.
Let $A$ be an integer non-negative matrix (every entry of $A$ is non-negative) and $x = (x_1,...,x_n)^T$ the probability Perron-Frobenius ...
- 351
5
votes
3
answers
739
views
Smoothness of hyperplane sections
Suppose $X\subset \mathbb{P}^n$ is a smooth hypersurface defined over $\mathbb{Q}$. For a "generic" prime $p$, what can be said about the set of hyperplanes $H$ in $\mathbb{P}^n(\mathbb{F}_p)$ for ...
- 13.1k
5
votes
3
answers
1k
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Proof that domains of positivity of symmetric nondegenerate bilinear forms are self-dual cones?
Max Koecher (for example, in The Minnesota Notes on Jordan Algebras and Their Applications; new edition: Springer Lecture Notes in Mathematics, number 1710, 1999), defined a domain of positivity for a ...
47
votes
4
answers
7k
views
Using linear algebra to classify vector bundles over ℙ¹
There is a theorem of Grothendieck stating that a vector bundle of rank $r$ over the projective line $\mathbb{P}^1$ can be decomposed into $r$ line bundles uniquely up to isomorphism. If we let $\...
- 533
14
votes
2
answers
2k
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Semi-linear operators
If $V_1$ and $V_2$ are finite-dimensional vector spaces over a field $E$, each equipped with an $E$-linear operator $\phi$, we can tell if $V_1$ and $V_2$ are isomorphic as $\phi$-modules by comparing ...
- 1,680
5
votes
2
answers
752
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Is there a name for this algebraic structure?
I found myself "naturally" dealing with an object of this form:
X is a complex vector space, with a "product" (a,b) → {aba} which is quadratic in the first variable, linear in the second, and ...
- 971
7
votes
4
answers
736
views
Simply connected quasi-projective varieties in positive characteristic
I am looking for examples of non-projective (quasi-projective) varieties $X$ defined over a field of positive characteristic, which have trivial étale fundamental group.
It is well known that the ...
- 4,450
22
votes
3
answers
2k
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One dimensional (phi,Gamma)-modules in char p
I would like to better understand the simplest case of the correspondence between Galois representations and (phi,Gamma)-modules. Namely, consider 1-dimensional Galois representations of $G_{Q_p}$ ...
- 1,680
9
votes
1
answer
763
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Restriction theorems over finite fields
A short while ago, Dvir proved the Kakeya conjecture over finite fields. Does this have any implications for restriction theorems over finite fields? I am aware only of implications going in the ...
- 20.2k
19
votes
1
answer
2k
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The Infinitesimal topos in positive characteristic
This question was inspired by and is somewhat related to this question.
In his article "Crystals and the de Rham cohomology of schemes" in the collection "Dix exposes sur la cohomologie ...
- 4,450
6
votes
2
answers
507
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Concerning the dimension of a complex variety modulo a prime
Let V be a complex affine variety given as the vanishing set of a set of polynomials with integral coefficients. I have 3 questions.
1)
Under what assumption will the dimension of V over C remain ...
- 63
4
votes
3
answers
763
views
Non-affine, projective vector field on $\mathbb{R}^n$
I wanted recently to discuss with a fairly elementary mathematics class the kinds of self-maps of Euclidean space that carry triangles to triangles. Obviously linear maps do this, and it seemed just ...
1
vote
2
answers
540
views
Using Wavelet Transforms to Approximate Matrices
It's a long time since I worked on this kind of problem, so please bear with me.
I have an approximate inverse matrix that I'm using as a preconditioner to solve the conjugate gradient method. ...
- 221
4
votes
2
answers
923
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 ...
- 9,132
9
votes
2
answers
984
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Spheres over rational numbers and other fields
Let K be an ordered field. Define the n-sphere:
$$S^n(K) := \{ (x_1,x_2,\dots,x_n+1) \in K^{n+1} \mid \sum_{i=1}^{n+1} x_i^2 = 1 \}$$
A set of vectors $v_1, v_2, \dots, v_r \in S^n(K)$ is ...
- 7,320
6
votes
2
answers
364
views
Algebraic characterization of transitive spaces of matrices
Fix an integer $d \ge 2$ and let $M_d$ be the space of real $d \times d$ matrices. Let $E$ be a vector subspace of $M_d$. We say that $E$ is transitive if $E \cdot \mathbb{R}^d_* = \mathbb{R}^d$, ...
- 2,479
42
votes
6
answers
12k
views
A slick proof of the Bruhat Decomposition for GL_n(k)?
On one of my exams last year, we were given a problem (we chose five or six out of eight problems) on an exam, the goal of which was to prove the Bruhat decomposition for $GL_n(k)$. I was one of the ...
- 19.6k