Skip to main content

All Questions

1,907 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
5 votes
0 answers
504 views

How to compute the volume of a region transformed by a matrix?

This is a rewrite of the OP's question to emphasize what I think are the research level issues here. Let $\mathscr{R}$ be a bounded convex body in $\mathbb{R}^n$ and let $H : \mathbb{R}^n \to \mathbb{...
RyanChan's user avatar
  • 550
5 votes
0 answers
436 views

Non-diagonalizable positive matrices

Let $n\geq 3$ and $E_n$ be the set of $n\times n$ matrices $A$ satisfying the $3$ following properties: $\bullet$ its entries $(a_{i,j})$ are positive integers. $\bullet$ the eigenvalues of $A$ are ...
loup blanc's user avatar
  • 3,741
5 votes
0 answers
91 views

elementary matrices over a regular ring

Let $n\in\mathbb{N}$ with $n\geq 3$. Let $A$ be a regular ring and $\gamma\in GL_{n}(A)$. We suppose that $\gamma$ is homotopic to the identity, i.e. there exists $\alpha\in GL_{n}(A[X])$ such that $\...
prochet's user avatar
  • 3,472
5 votes
0 answers
372 views

Gelfand pairs in $SO(p,q)$

I am considering the groups $SO(p,q)$ over the reals. And inside it some parabolic subgroup, $P$. It can be the minimal parabolic but that is not the issue. It is well known that $P$ has a Levi ...
Gal Yehoshua's user avatar
5 votes
0 answers
125 views

Low-rank approximation over finite fields

Consider the finite field $\mathbb{F}_p$ with prime $p$. Let $I=\{ 0,1,2,...,|I|-1 \}\subset \mathbb{F}_p$ be an "interval". What can be the largest size $|I|$, such that there exists a $2\times 2$ ...
fil's user avatar
  • 51
5 votes
0 answers
161 views

Where have you encountered "arrangement spaces"?

I am compiling a paper in which I advertise (and use) the following notion of arrangement spaces (I made up the name, as I found no standard name in the literature). Let $v_i\in\Bbb R^d,i\in N:=\{1,.....
M. Winter's user avatar
  • 13.6k
5 votes
0 answers
350 views

How to calculate the volume of a parallelepiped in a normed space?

Let $E$ be a real normed space, and let $v_1,...,v_n\in E$ be linearly independent. The parallelepiped defined by these vectors is $P=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Since $E$ is a ...
erz's user avatar
  • 5,529
5 votes
0 answers
150 views

monomer-dimer tiling of a Young diagram

As a modest start, I propose the below problem for a special set of partitions. Perhaps it is known. Let $\lambda_n=(n,n-1,\dots,2,1)$ be the staircase partition and its corresponding Young diagram $...
T. Amdeberhan's user avatar
5 votes
0 answers
357 views

$\text{Determinant}=(\sum \text{Determinant})^2$

Denote by $\delta_{n-1}=(n-1,n-2,\dots,1,0,0,\dots)$ the staircase partition and the embedded partition $\lambda=(\lambda_1,\lambda_2,\dots)\subset\delta_{n-1}$. QUESTION 1. Is this true? $$\det\...
T. Amdeberhan's user avatar
5 votes
0 answers
291 views

Can we represent partitions by mutually parallel lines in the plane?

Lately I have become interested in the following idea: Suppose $n$ is a positive integer and $[n]=\{1,2,3,...,n\}$. Suppose we have 3 distinct partitions $b$, $g$, and $r$ of $[n]$. Assume that the ...
David Richter's user avatar
5 votes
0 answers
2k views

What is the Jarlskog invariant, conceptually?

Let $U$ be a $3\times 3$ unitary matrix, and call $(u_{ij})$ its coefficients. For $i,j,k,\ell$ in $\{1,2,3\}$ with $i\neq j$ and $k\neq\ell$, consider the quantity: $$J_{ij,k\ell} := \operatorname{...
Gro-Tsen's user avatar
  • 32.5k
5 votes
0 answers
440 views

Why does Faltings' Siegel lemma imply Siegel lemma?

Recall the Siegel lemma: Let $A = (a_{ij})$ be an $N \times M$ matrix with rational integer coefficients. Put $a = \max_{i,j} |a_{ij}|$. Then, if $N < M$, the equation $Ax = 0$ has a solution $x\in\...
joaopa's user avatar
  • 3,996
5 votes
0 answers
397 views

spectrum of orthogonality graphs

The orthogonality graph $\Omega(n)$ with $2^n$ vertices is the graph with vertex set $\{-1,+1\}^n$, with two vertices being adjacent if and only if they are orthogonal (as vectors in the standard ...
Clive elphick's user avatar
5 votes
0 answers
162 views

Homogeneous linear and quadratic inequalities

I have a bunch of vectors $b_i \in R^n$ for $i = 1,\ldots,N$ and a bunch of (indefinite) matrices $A_j$ for $j = 1,\ldots,M$. Let's consider the set $S \subset R^n$ of $x \in R^n$ vectors such that $$...
Fetchinson0234's user avatar
5 votes
0 answers
856 views

Dual of representation is irreducible implies the representation is irreducible?

Suppose $V$ is an infinite dimensional $\mathbb{Q}_p$-representation of Lie algebra $\mathfrak{g}$ over $\mathbb{Q}_p$. If its dual representation $V^{\prime}$ is irreducible then, is it always true ...
MathStudent's user avatar
5 votes
0 answers
586 views

A minimal eigenvalue inequality

Suppose $A$ is an $n\times n$ real symmetric positive definite matrix. Let $A^{(-1)}_{i,j}$ be the $n\times n$ matrix the entries $(i,i),\,(i,j),\,(j,i),\,(j,j)$ of which equal to the corresponding ...
Hans's user avatar
  • 2,239
5 votes
0 answers
1k views

a new proof of a square matrix is similar to its transpose (not completed)

I know there exist a classical way to prove that a square matrix is similar to its transpose in $C$ using jordan matrices. I would like to prove it in another way but i have some difficulities to ...
SC_thesard's user avatar
5 votes
0 answers
137 views

How many CNOT gates are needed to compute scalar multiplication in a finite field of characteristic 2?

I want to know if scalar multiplication over a finite field of characteristic 2 is generally easier to compute than a linear transformation over a vector space over $F_{2}$. Suppose that $p(x)$ is an ...
Joseph Van Name's user avatar
5 votes
0 answers
80 views

Generalized rational form

Fix a field $k$. It is well known that any square matrix is similar to a block-diagonal matrix such that every block is of the following form: $1$ in the subdiagonal, a bunch of numbers in the last ...
Bissaque's user avatar
5 votes
0 answers
315 views

Is there a matrix with this specific quadratic determinant?

We have $\det M=(a+b)(c+d)$ where $M=\begin{bmatrix} a& 0& -1& 0\\ 0& c& 0& -1\\ b& 0& 1& 0\\ 0& d& 0& 1 \end{bmatrix}$ and $\det M'=(a'+b')(c'+d')$...
Turbo's user avatar
  • 13.9k
5 votes
0 answers
330 views

Best Approximation in Operator/non-Frobenius Norm

Since the Frobenius norm on matrices is generated by an inner product, solving the optimization/approximation problem of approximating an operator $X$ with a scalar multiple of another operator $Y$ $$\...
Conner DiPaolo's user avatar
5 votes
0 answers
743 views

Is this set empty?

Suppose we have two rank $n-1$ matrices in $\Bbb Z^{(n-1)\times n}$ given by $$C=\begin{bmatrix} c_{1}& -d_{1}& 0& 0&\dots 0& 0\\ 0& c_{2}& -d_{2}& 0&...
Turbo's user avatar
  • 13.9k
5 votes
0 answers
235 views

Riemann theta function inequality for a class of large random matrices

The following is essentially the same question as in this previous post, but since I have completely re-formulated it (hopefully for the better ;-), I decided to post a new question instead of an edit....
Dierk Bormann's user avatar
5 votes
0 answers
240 views

Complexity of approximating the size of the range of a matrix

Given an $m$ by $n$ matrix $M$ with $m \leq n$ and elements from $\{-1,1\}$, let us define: $$S_M = |\{Mx : x \in \{-1,1\}^n\}.$$ It is NP-hard to compute $S_M$ exactly I believe by applying the ...
Simd's user avatar
  • 3,377
5 votes
0 answers
234 views

A combinatorial problem

What is the largest $m\times m$ $0/1$ matrix of real rank $n$ with every square submatrix sized at least ${n^{\gamma}}\times{n^{\gamma}}$ distinct for some fixed $\gamma>0$? Upper Bounds: Number ...
user avatar
5 votes
0 answers
216 views

Existence or construction of a sequence of orthogonal matrices with three properties

This is a problem that I encountered during my research, and I have spent a good amount of time on it without success. So I am reaching out for help .... Any pointers or suggestions are appreicated! ...
Chee's user avatar
  • 984
5 votes
0 answers
251 views

Eigenvalues of a certain product of matrices with special structure

Let $d$ and $c$ be positive integers and $q = dc$. Let $G$ be a $q$-by-$q$ positive semi-definite real matrix with eigenvalues all $\le 1$, and define the $q$-by-$2q$ matrix $A = [G\hspace{1em}\...
dohmatob's user avatar
  • 6,853
5 votes
0 answers
135 views

Relative invariants of $P\oplus P^*$

Let $P$ be a $\mathrm{GL}(V)$-module, and assume that the decomposition of $P$ into irreducible submodules is known. By a relative invariant of the module $P\oplus P^*$, I mean a homogeneous nonzero ...
Giovanni Moreno's user avatar
5 votes
0 answers
310 views

Biggest (or large) rectangle in a polytope

I need an efficient method to construct a (hyper)rectangle inside a polytope with a lot of dimensions (say $100 < d < 1000$). Ideally I'd want the biggest possible rectangle, but as I don't ...
Elliot Gorokhovsky's user avatar
5 votes
0 answers
187 views

Mapping the standard $(n!-1)$-simplex into $GL_n(\mathbb C)$

Let $\Delta^{n!-1}$ be the standard $(n!-1)$-simplex whose vertices are indexed by the elements of the symmetric group $S_n$, that is \begin{equation} \Delta^{n!-1} = \left\{ (t_\sigma)_{\sigma\in S_n}...
Chris Ramsey's user avatar
  • 3,984
5 votes
0 answers
376 views

Non-linear positive map

In the paper titled "Nonlinear completely positive maps" M. D. Choi and T. Ando extended natural definition of completely positive maps ignoring the linearity condition (Aspects of positivity in ...
RSG's user avatar
  • 421
5 votes
0 answers
254 views

A weak Perron-Frobenius property for sets of positive matrices

A popular form of the Perron-Frobenius theorem states the following result: if $A$ is a $d \times d$ real matrix all of whose entries are positive, then the spectral radius of $A$ is a simple ...
Ian Morris's user avatar
  • 6,206
5 votes
0 answers
620 views

Is there a method to simultaneously block-diagonalize a set of group matrices?

Assume that you are explicitly given the representation matrices of a group. How does one go about finding that common basis which will find the irreducible components of all of them simultaneously? ...
user6818's user avatar
  • 1,893
5 votes
0 answers
167 views

A specific case of the $p$-center problem

Given a fixed positive integer $m$, let $\cal{S}$ be the subset from $\mathbb{R}^m$ defined as $\cal{S} = \{u \in \mathbb{R}^m \mid \forall i \in \{1, \dots, m\}, u(i) > 0$ and $\sum_{i=1}^m{u(i) = ...
user109711's user avatar
5 votes
0 answers
255 views

Existence of a matrix product from its eigenvalues

Let A and B be two positive definite, real, symmetric matrices. The eigenvalues of A, B and AB, denoted by $\lambda(X)$, obey the relation (from Bhatia): $$ \lambda^\downarrow(A) \cdot \lambda^\...
ScienceSnake's user avatar
5 votes
0 answers
428 views

Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course. Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
yang's user avatar
  • 181
5 votes
0 answers
2k views

A stronger Cauchy-Schwarz inequality for traces of compression matrices

Assume that $A$ and $B$ are contractions, so $I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let $C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that: $$Tr\left(\frac{1}{1-AA^T}\right)...
math110's user avatar
  • 4,270
5 votes
0 answers
235 views

A question on hyperplanes in partial linear spaces and hypergraphs

A partial linear space (or a linear hypergraph) is a point line geometry $(P,L,I)$ where for every pair of points there is at most one line incident with both of them. A hyperplane in a partial linear ...
Anurag's user avatar
  • 1,197
5 votes
0 answers
604 views

The twisted kiss of the curvaceous cubic and the staid tetrahedron (references)

(Migrated from MSE) While investigating some operators, I came across some relations between the twisted cubic curve and the tetrahedron that link together some notions in differential geometry, ...
Tom Copeland's user avatar
  • 10.5k
5 votes
0 answers
148 views

Groups of operators between local unitaries and full unitaries

Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...
Nathaniel Johnston's user avatar
5 votes
0 answers
194 views

A linear optimization problem on a graph

Let $G=(V,E)$ be a finite graph and let $f$ be any positive function defined on the vertices. Put weights on the vertices $v_{i}$, way $w_{i}$ so that $\sum_{i=1}^{n}w_{i}\leq 1$. Assume that every ...
TOM's user avatar
  • 2,288
5 votes
0 answers
231 views

Singularity of an $l\times l$ matrix whose entries are $2l$-th roots of unity

Let $l$ be a positive integer, $\zeta$ be a primitive $2l$-th root of unity in $\mathbb{C}$, and $\alpha,\beta$ be $\pm1$ sequences of length $l$, i.e. $\alpha_k=\pm1,\beta_k=\pm1$ for $k=0,\dots,l-1$....
Binzhou Xia's user avatar
5 votes
0 answers
392 views

Preconditioner for finding the smallest eigenpairs of a large, but structured, matrix

I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $L$. $L$ is a graph Laplacian, with the following structure: $L = D -...
Jeff's user avatar
  • 500
5 votes
0 answers
112 views

Inverses of the sums of all possible subsets of a set of symmetric and positive definite matrices

I have a set of $c$ matrices $A_1 ... A_c$ which are all symmetric and positive definite. I would like to calculate the inverses of all the possible sums, i.e. $(A_1+A_2)^{-1},(A_1+A_3)^{-1},(A_2+A_3)...
Tzonathan's user avatar
5 votes
0 answers
153 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 ...
Theo Johnson-Freyd's user avatar
5 votes
0 answers
406 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 ...
gondolf's user avatar
  • 1,503
5 votes
0 answers
1k views

Tensors as multilinear maps

I am aware that many books on differential geometry define tensors as multilinear maps. Namely $$ V\otimes W := L_2(V^* \times W^*,\Bbb F) $$ I am also aware that this space is isomorphic to the ...
Markus's user avatar
  • 61
5 votes
0 answers
596 views

Literature on Exponential of a Quadratic Form

Let $A_i$, $i=1,\dots,L$ be given $N\times N$ positive definite real matrices. I have this sum of exponentials \begin{align} f(\mathbf{x})=\sum_{i=1}^{L}\operatorname{exp}(-{\mathbf{x}^T\mathbf{A}_i\...
dineshdileep's user avatar
  • 1,421
5 votes
0 answers
442 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 ...
Alexander Shamov's user avatar
5 votes
0 answers
160 views

reference for perturbation of projection result

Let $A$ and $B$ have the same rank and dimensions. If $P_A$ denotes the projection onto the range space of $A$, then $$ \|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2). $$ ...
AatG's user avatar
  • 922

1
3 4
5
6 7
39