Skip to main content

All Questions

1,809 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
0 votes
0 answers
35 views

What is the locus defined by those equations?

I would like to know what is the locus of $x \in \Bbb R_+^n$ ($n=2$ would already be fine) defined by $\sum a_i \cdot x_i$ s.t. $a_i+\epsilon \geq 0$, $\epsilon \in \Bbb R$. I know that if $\...
MysteryGuy's user avatar
0 votes
0 answers
58 views

Quadrics over the univariate function field with discriminant of minimal degree

Consider a non-degenerate quadric $Q(x,y,z) \subset \mathrm{P}^2$ over the univariate function field $\mathbb{F}_p(t)$, where $\mathbb{F}_p$ is a prime finite field, $p > 2$. For simplicity assume ...
Dimitri Koshelev's user avatar
0 votes
0 answers
50 views

Examples of Binary Functions that Yield Regular Graphs with Invertible Adjacency Matrix

Question: What are, provided their existence, examples of functions $f$ with the following properties: \begin{align}f:& \ \mathbb{N}\times\mathbb{N}\ni(i,j)&\mapsto\ \quad\quad\...
Manfred Weis's user avatar
  • 13.2k
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 ...
Dattier's user avatar
  • 4,074
0 votes
0 answers
64 views

Probability of collision of sums of vectors

Let $S_1$ and $S_2$ be sets of vectors from $\mathbb{R}^d$ that are distinct and let $\sigma(\cdot)$ be a non-linearity, e.g., a componentwise sigmoid function. Does there exist a random matrix $R \...
Christopher's user avatar
0 votes
0 answers
369 views

Finding a point in the relative interior of the convex hull of a set of integer-valued vectors

Let $X \subset \mathbb{Z}^n$ be the set of integer-valued vectors satisfying a system of linear constraints. We can suppose that $X$ is the set of integral points in a given polyhydral set $Y \subset \...
rasul's user avatar
  • 136
0 votes
0 answers
224 views

Upper bound on matrix perturbation such that all eigenvalues lie within the unit circle

Consider the matrix $$N=\left[\matrix{\mathbb{I}_n-\epsilon L & X\\ \epsilon Y & Z}\right]$$ where $\epsilon>0$ is a small positive parameter and $Z$ is a square $m\times m$ matrix with ...
CTNT's user avatar
  • 101
0 votes
0 answers
181 views

Number of Symmetric matrices

Let $S_m(q)$ denote the space of all $m\times m$ symmetric matrices over the finite field $\mathbb{F}_q$ of size $q$. What is the number of matrices $A=(a_{ij})\in S_m(q)$ of rank at most $3$ and $a_{...
Singh's user avatar
  • 179
0 votes
0 answers
184 views

Oja's rule gives unit eigenvectors

Does Oja's rule for normalized Hebbian learning always result in a unit eigenvector which corresponds to the largest eigenvalue? Or are there any specific conditions or assumptions under which this is ...
S_S's user avatar
  • 101
0 votes
0 answers
98 views

Eigenvalues of a sequence of matrices involving the divisor function

Let $A_{n,k},k=1,\ldots,n$ be a sequence of $n\times n$ upper triangular matrices where $A_{n,1}=I_n$ and $A_{n,k},\quad 2\leq k\leq n$ be a regularly shifted and scaled matrix, with $P_{n,k}$ an $n\...
kodlu's user avatar
  • 10.4k
0 votes
0 answers
282 views

A symmetric matrix with nonzero principal minors is cogredient to a diagonal matrix via an upper triangular

A paper I'm reading in representation theory states the following result: Let $F$ be a field of characteristic zero, and $x$ a symmetric matrix in $M_n(F)$ all of whose principal minors are not zero. ...
D_S's user avatar
  • 6,180
0 votes
0 answers
93 views

Changing Couplings of Discrete Random Variables

Let $X,Y$ be two discrete random variables. Two joint mass distributions (couplings) with marginals $X$ and $Y$ and with entries $p_{i,j}=\mathbb{P}_1(X=i,Y=j)$ and $p_{i,j}'=\mathbb{P}_2({X=i,Y=j})$ ...
The Substitute's user avatar
0 votes
0 answers
100 views

Solutions of the linear equation from K[[X_1,X_2,X_3]] to K[[X_1,X_2]]

Let $A_3 := K[[X_1,X_2,X_3]]$ be a three-variable formal power series ring over a field $K$. We consider a linear equation $(\sharp) \phantom{aa} a_1(X_1,X_2,X_3)Y_1 + \ldots + a_n(X_1,X_2,X_3)Y_n = ...
Pierre's user avatar
  • 563
0 votes
0 answers
290 views

Need any information about an affine lattice

Motivation - I was thinking about calculating the integrals from An interesting integral expression for $\pi^n$? using old plain Riemann sums. There, one needs integrating over that part of $[0,1]^n$ ...
მამუკა ჯიბლაძე's user avatar
0 votes
0 answers
40 views

Question about a systematic row reduction algorithm for compressive sensing

Suppose a ''brute-force'' algorithm is designed to systematically select from the first $n$ columns of an $m \times (n+1),$ $m<n$ augmented matrix $G$ representing a consistent underdetermined ...
Thomas Rasberry's user avatar
0 votes
0 answers
89 views

Show that a certain ratio of diagonal entries dominates a certain ratio of singular values

Let $D=[d_{ij}]_{i,j=1,\ldots,4}$ be a $4 \times 4$ “density matrix”, that is, a Hermitian (possibly symmetric) positive definite matrix having trace 1—that is, the (nonnegative) diagonal entries sum ...
Paul B. Slater's user avatar
0 votes
0 answers
263 views

Does AX+XA=0 have any non-trivial solutions?

Let $X$ be a continuous linear self-adjoint operator on some Hilbert space $H$ and for arbitrary compact operators $A$ we have: $XA+AX=0.$ Does this imply that $X=0$ or can there be non-trivial ...
Kinzlin's user avatar
  • 305
0 votes
0 answers
84 views

Number of solutions to non-linear equations

As part of our project, we are required to determine the total number of distinct solutions to the following equations. There are $n$ variables of one type, say $\{p_i\}_{i=1}^n$, $m$ variables of ...
user104452's user avatar
0 votes
0 answers
68 views

A seemingly easy integer programming question

Let $k, m \in \mathbb{Z}_{ > 1}$. Let $a \in \mathbb{Z}_{> 0}^m$ and $t \in \mathbb{Z}^k$. Let $\varepsilon = (\varepsilon_{i,j})_{1 \leq i \leq m \\1 \leq j \leq k}$ be a matrix with entries in ...
Alex's user avatar
  • 501
0 votes
0 answers
69 views

Quasi-stationary measure on a finite graph equals stationary measure?

Assume the simple random walk $X$ on the graph $G(V,E)$, s.t. $G$ is simple, undirected, finite, connected and let $B \subset V$, s.t. $V\setminus B$ is connected. Let $\sigma_B$ be the quasi-...
jondal's user avatar
  • 71
0 votes
0 answers
336 views

Pfaffian minors of skew symmetric matrix under perturbation

Suppose $A$ be a skew-symmetric matrix whose entries are positive numbers. A perturbation of $A$, $A'$, is obtained by adding another skew-symmetric matrix whose entries are positive integers. My ...
SiOn's user avatar
  • 493
0 votes
0 answers
202 views

Conical combination of rank 1 matrices with nonnegative entries

Let $A = (a_{ij})$ be an $n \times n$ matrix with entries in the nonnegative real numbers $\mathbb{R}_+$. Suppose that, for each $i = 1,\ldots, n$, the sum $b_i := a_{i1} + \cdots + a_{in}$ of the ...
user avatar
0 votes
0 answers
182 views

The minimal angle between a vector and a subspace, given another minimal angle

Let $X$ be a set of vectors in $\mathbb{R}^d$. Denote $\theta$ to be the minimal angle between any two vectors in $X$. Denote $\alpha$ to be an angle between (1) some vector $y\in X$ and (2) some ...
Daniel Soudry's user avatar
0 votes
0 answers
166 views

Endomorphism ring as ind-pro object

Let $Vect_f$ be the category of finite-dimensional vector spaces. This category comes with a very well-behaved duality functor. Now the ind-completion of this category (if I understand correctly) is ...
Dmitry Vaintrob's user avatar
0 votes
0 answers
55 views

Continuous Functions On Grassmannans under containment restrictions

Let $V$ be a vector space. Suppose that for a $x\in V$, we are given some subspace of dimension no more than d (e.g., the kernel of some operator defined on V, which varies smoothly with x), call it $\...
Miel Sharf's user avatar
0 votes
0 answers
698 views

Singular Values of Linearly Combined Matrices

I have a question related with singular values of matrix sums. Let's assume I have matrices $A$, $B$, and $D$ (positive, semi-definite) where $D = A + B$. For singular values of $D$, I know that $$ ...
ciyo's user avatar
  • 101
0 votes
0 answers
182 views

How to understand the change of basis in a certain differential equation?

Let $w:\mathbb{R} \times \mathbb{R}^n \rightarrow Mat(n,\mathbb{R})$ be a smooth function, $R_{ij}$ be a fixed skew-symmetric $n\times n$ real matrix, and $A\in\mathbb{R}$. Consider the equation $$\...
geometricK's user avatar
  • 1,903
0 votes
0 answers
185 views

Sum of unit vectors always has a binary span after constrained permutations

Conjecture: Let $e_1 = (1,0,\ldots,0), \ldots , e_{m_1+m_2} = (0,\ldots,0,1)$ be the unit vectors of the standard basis $E$ of $\mathbb{R}^{m_1+m_2}$. An enumeration $ E \cup -E = \{f_1, \ldots, ...
user95393's user avatar
  • 121
0 votes
0 answers
111 views

Sandwich rule for Lie algebras

On an infinite dimensional vector space an operator can be onto but not one-to-one (and vice versa). This arises the following question. Let $L_1$ and $L_2$ be Lie algebras (infinite dimensional, over ...
Bedovlat's user avatar
  • 1,959
0 votes
0 answers
252 views

Hadamard product (Schur product) in $L^2[0,1]$

Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...
Obriareos's user avatar
  • 195
0 votes
0 answers
838 views

Simplifying product of matrix exponential?

Is there a known generalization for n-term matrix exponential multiplication? I am aware that the Baker–Campbell–Hausdorff formula could be used, e.g.: ...
norok2's user avatar
  • 111
0 votes
0 answers
114 views

A linear combination problem

Given $0/1$ $n\times n$ matrix $M$. Suppose we have $2$ vectors $\lambda,\mu\in\Bbb R^{1\times n}$ such that both $$\lambda M\in\{0,1\}^{1\times n}$$ $$M\mu'\in\{0,1\}^{n\times 1}$$ holds with $'$ ...
user avatar
0 votes
0 answers
322 views

Comparison of Parameter estimation using maximum likelihood and Maximum entropy

I am not sure if the question is appropriate but I want to try my luck. One can estimate a parameter using maximum likelihood and we know it is optimal. On the other hand there are methods which uses ...
Creator's user avatar
  • 495
0 votes
0 answers
212 views

Can we drop commutativity assumption?

Let $A$ be an associative algebra with a unit over a field $k$. fix $n > 1$. Define a $k$-algebra structure on the vector space $A^{\otimes n} = A \otimes_k \dots \otimes_k A$ (where there are $n$ ...
Ed Fischer's user avatar
0 votes
0 answers
84 views

Can we increase spectral norms of All maximum size square submatrices by orthogonal perturbation?

Let the matrix $A$ consist of $k$ columns from some $n \times n$ orthogonal (unitary) matrix. It is obvious that there is no perturbation of $A$ which leaves its columns orthonormal, increases ...
Yura's user avatar
  • 1
0 votes
0 answers
53 views

Largest Set of Special Unitary Matricies With Invariant Subspace For Adjoint Action

I am trying to solve the following. Given the special unitary group $SU(n)$ and its adjoint action $Ad_{U}: \mathfrak{su}(n) \rightarrow \mathfrak{su}(n)$, what is the largest subset of $SU(n)$ such ...
Benjamin's user avatar
  • 2,099
0 votes
0 answers
369 views

Bounds on the smallest eigenvalue of a Hankel matrix

Let $H=H_n$ be a positive definite Hankel matrix of size $n$ with $\lambda_n$ is it's smallest eigenvalue. What bounds are known on $\lambda_n$ in terms of the entries on $H$. I can see some results ...
gil's user avatar
  • 265
0 votes
0 answers
89 views

Degree of permutation of hypercube

Given $S_0\cup S_1=T_0\cup T_1=\{0,1\}^n$, $S_0\cap S_1=T_0\cap T_1=\emptyset$, with $|S_i|=|T_i|$ for both $i\in\{0,1\}$, what is degree of transformation that simultaneously maps $S_i$ to $T_i$ for ...
Turbo's user avatar
  • 13.9k
0 votes
0 answers
614 views

Hadamard / matrix product adjoint

First of all I would like to thank everyone over here at mathoverflow for their amazing generosity and help (for both pros and newbies like myself). I apologize if this question seems dumb; I'm a new ...
M.Salem's user avatar
0 votes
0 answers
89 views

Decomposing matrices to lower ranks

Given real matrix $M\in\{0,1\}^{n\times n}$ of rank $r$. How many $M_i\in\{0,1\}^{n\times n}$ of rank $s$ does one need to write $M'=\sum_{i=1}^ta_iM_i$ for some $a_i\in\Bbb R$ where maximum absolute ...
Turbo's user avatar
  • 13.9k
0 votes
0 answers
83 views

Bits of precision matrix reconstruction

We have a real rank $r$ matrix $M\in\{0,1\}^{n\times n}$. Suppose we have diagonalized using $LMR=D$. I want to recover a real matrix $\widetilde{M}$ such that maximum absolute entry of $\widetilde{...
Turbo's user avatar
  • 13.9k
0 votes
0 answers
161 views

Way to parameterise sparse multi diagonal matrix

I have an NxN matrix S that looks like this: $$ S^{-1} = K^{-1} + \Lambda $$ where N is a multiple of 3, both K and S are positive definite matrices, and Lambda is $$ \Lambda = \begin{bmatrix} x &...
tbi's user avatar
  • 1
0 votes
0 answers
345 views

Shift functor and the origin of the linear decalage isomorphism

Let $(\mathbf{Vec}_\mathbb{Z}(\mathbb{K}),\otimes,\tau)$ be the symmetric monoidal category of $\mathbb{Z}$-graded $\mathbb{K}$-vector spaces, where $\otimes$ is the tensor product of $\mathbb{Z}$-...
Mark.Neuhaus's user avatar
  • 2,074
0 votes
0 answers
182 views

Question about majorization of eigenvalues after conjugation

Let $A$ and $B$ be $n \times n$ positive semidefinite matrices with eigenvalues $\alpha_1 \ge \alpha_2 \ge \ldots \ge \alpha_n$ and $\beta_1 \ge \beta_2 \ge \ldots \ge \beta_n$ respectively. ...
David Roberson's user avatar
0 votes
0 answers
227 views

Canonical forms of symmetric/skewsymmetric quaternionic matrix

$A$ belongs to $n$-dimensional quaternion symmetric matrix, in the sense that $A=A^T$, where $T$ means transpose. Under transformation $U$, $A\rightarrow U\cdot A\cdot U^T$, where $U$ is $n$-dim ...
Shenghan Jiang's user avatar
0 votes
0 answers
68 views

Estimate bounds on Minkowski distance from point to a segment in Lp space

Assumptions Let $L_p(x,y)=(\sum_i|x_i - y_i|^p)^{1/p}$ (Minkowski metric), $a,b$ be arbitrary $n$-dimensional points, $c$ be a point that satisfies $L_p(a,b) = L_p(a,c) + L_p(c,b)$, i.e., a point ...
tomash's user avatar
  • 101
0 votes
0 answers
163 views

Applying a linear operator to a basis set following SVD orthonormalization

Define $\Phi$ as an $N$x$N$ dense, symmetric matrix, who's columns represent a set of $N$ non-orthogonal bases. My intention is to: decompose $\Phi$ via SVD: $U \Lambda V^T = \Phi$ to create it's ...
Greyman's user avatar
0 votes
0 answers
145 views

Does there exist this special kind of homeomorphism?

Let $A,B\subset\mathbb{R}^n, n\geq 2,$ are two different shaped spindles. One is thick and one is thin. (Sorry for my unprofessional statements. Unsure about how to say it rigorously.) So there are ...
Lucy's user avatar
  • 183
0 votes
0 answers
80 views

Finding gradient of an optimization

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me? Assume that we have an optimization ...
Royeh's user avatar
  • 161
0 votes
0 answers
81 views

A Optimization problem using co-ordinates of joint numerical range.

Let $\mathbf{A}_1,\dots,\mathbf{A}_L$ be $N\times N$ hermitian matrices. Define the mapping from the $N-$dimensional unit sphere to $\mathbb{R}^L$ as \begin{align} \mathcal{S}=\{\left(\mathbf{u}^H\...
dineshdileep's user avatar
  • 1,421