All Questions
3,680 questions
2
votes
1
answer
200
views
Upper bound for the rank of a Gram-type matrix
Let $V = (v_{1},...,v_{N})$ and $W = (w_{1},...,w_{N})$ be 2 sets, each containing $N$ vectors from $\mathbb{R}^{n}$; i.e, $v_{j}, w_{j} \in \mathbb{R}^{n}$ for all $1 \leq j \leq N$. Assume that $N$ ...
- 93
0
votes
1
answer
744
views
A RKHS interpretation of the Rydberg formula for hydrogen and an application for physics?
I was thinking if it is possible to define an inner product between two small physical objects with a positive definite kernel and was led to look at the Rydberg formula:
The Rydberg formula for ...
- 3,054
2
votes
1
answer
298
views
Is there a combinatorial interpretation for the change of basis matrix in the Frobenius normal form representation?
Let $G$ be a graph on $n$ vertices. Let $A$ be the adjacency matrix of $G$ (i.e., rows and columns of $A$ are indexed by vertices of $G$, and the $(v,w)$ entry of $A$ is $1$ if $(v,w)$ is an edge in $...
- 557
1
vote
0
answers
134
views
Matrix valued word embeddings for natural language processing
In natural language processing, an area of machine learning, one would like to represent words as objects that can easily be understood and manipulated using machine learning. A word embedding is a ...
1
vote
0
answers
94
views
Linear Program Optimal Value
If $f(A,b,c)$ is the optimal value of a linear program
$\min c.x$
subject to $A.x \leq b ; x \geq 0.$
Does $f(A,b,c)$ have a piecewise polynomial/rational upper bound in $(A,b,c)$ on the domain of ...
2
votes
0
answers
114
views
What is the quantity $\sqrt{\frac{c^2+d^2}{a^2+b^2}}$ of a matrix with determinant one?
Suppose that $A \in \mathbb R^{2 \times 2}$ has determinant one,
$$
A = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
$$
I'm currently working on a problem where I obtained a condition on the ...
- 173
0
votes
1
answer
171
views
Prime index subgroups of $\langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle$ that is invariant under matrix $Q$
Let $Q $ be a matrix in $ \operatorname{GL}(2, \mathbb{Q}) $ and consider the group
$G = \langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle := \langle Q^{i}(v) \mid i \in \mathbb Z, v \in \...
- 823
0
votes
1
answer
114
views
Mixed integer program and continuous Diophantine approximation
Let $n\in\mathbb{N}$ such that $n\geq 2$ and let $0<r<1$ be a real number. We wish to solve the following problem.
$$\min_{(t,(z_j)_{j=2}^n) \in \mathbb{R}\times \mathbb{Z}^{n-1}} t$$
subject to ...
2
votes
1
answer
198
views
An inequality related to matrix trace
$$Tr(A|A^T U \Sigma U^T|) \leq Tr(AA^T U \Sigma U^T)$$ where $A \in \mathbb{R}^{m\times n}$ is a real rectangular matrix, $U \in \mathbb{R}^{n \times n}$ is a orthonormal matrix and $\Sigma$ is a ...
- 21
4
votes
1
answer
230
views
Conditions for distinct nonzero eigenvalues in product DAD for symmetric matrix A with repeated nonzero eigenvalues and diagonal matrix D
Let $A\in\mathbb{R}^{n\times n}$ be an real symmetric matrix with eigenvalues $\lambda_1, \lambda_2, \cdots, \lambda_n$ with some of which be nonzero and repeated, i.e., there exist $\lambda_i \ne 0$ ...
- 187
1
vote
1
answer
112
views
square matrix depending on complex value: spectral radius continous? [closed]
Let $A(z)$ be a $n\times n$ square matrix depending on the complex value $z$ and $\lambda_z$ is its spectral radius.
Is $\lambda_z$ continous or is it possible that it can jump? Or maybe someone knows ...
- 83
0
votes
0
answers
92
views
Norm of matrix product sum
Given matrices $A_{n\times n}, B_{n\times m}, C_{m\times m}$ such that $A^iBC^{N-i}$ is matrix with all zeros except upper right element for all $i$ from $0$ to $N$, what can we say about Frobenius ...
- 159
4
votes
1
answer
214
views
Diameter of the unimodular group with Gauss moves
$\DeclareMathOperator\GL{GL}$Consider the unimodular group $\GL_n(\mathbb{Z})$, consisting of integral matrices $A \in \mathbb{Z}^{n \times n}$ such that that $\det(A) =\pm 1$.
It is well known that ...
- 327
2
votes
0
answers
137
views
Decompose a rational matrix as an integer matrix and an inverse of integer matrix
Suppose we have a non-singular rational matrix $Q$, consider the the $\mathbb{Z}$-span of the columns of $Q$ and $Q^{-1}$, denote it as $H = {\rm Span}_{\mathbb Z} \{ Q(\mathbb Z^{n}), Q^{-1}(\...
- 823
1
vote
2
answers
137
views
Methods to solve for a matrix whose entries satisfy certain properties
(This question is a repost of a deleted question I asked, because the previous version had several elements missing)
Setting
For fixed $N \in \mathbb{N}$, I wish to compute the entries of a matrix $...
- 135
2
votes
1
answer
240
views
Is the problem of vertex enumeration from an H-representation of a polytope NP-hard?
According to the Wikipedia page on the issue, the vertex enumeration problem is NP-hard.
However, double description and reverse linear search are algorithms listed to solve the problem. Moreover, ...
- 123
1
vote
1
answer
519
views
Optimal transport between two matrices
I'm investigating the use of optimal transport to define a distance between non-negative matrices that satisfy the condition $\mathbf e^\top\mathbf M\mathbf e = 1$ (i.e., the sum of all elements in ...
- 243
1
vote
0
answers
35
views
Are there any known lower complexity bounds on solving positive semidefinite or positive semidefinite feasibility problems?
I've been trying to attack the problem posted here, about quickly checking if a matrix has any positive semidefinite completions. I suspect that the answer to the question is "no", because ...
- 111
5
votes
2
answers
508
views
Is there a name for this family of matrices?
Let $0<a_1<a_2<\cdots<a_n$ and let $A$ be the symmetric $n\times n$ matrix with
${ij}^\text{th}$ entry $A_{ij}=\min\{a_i,a_j\}$.
For example, if $a_i=i$ for each $i\le n=5$ then
$$A=\begin{...
- 51
1
vote
1
answer
425
views
How to represent infinite matrices in Mathematica?
I asked this question on Mathematica site and the answer was basically "it is impossible". So, some math theory needs to be thrown in besides coding.
Let us represent infinite matrices as ...
- 10.1k
2
votes
0
answers
51
views
What conclusions can I derive from this family of trace inequalities?
Problem. Let $n_1,\ldots,n_s,m_1,\ldots,m_s\ge 0$ be nonnegative integers and set $m := \sum_{i=1}^s m_i$ and $n := \sum_{i=1}^s n_i$. Let $\oplus$ be an operation on matrices which stacks them in a ...
- 283
1
vote
1
answer
252
views
Smith normal form and last invariant factor of certain matrices
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts can provide some relevant insight.
Suppose we have row vectors $x_1$, $x_2$ , $y_1$ , $y_2 ...
- 823
9
votes
1
answer
361
views
Conjugates iff conjugates over $\mathrm{GL}_n(\overline{\mathbb{F}_q})$?
Let $G$ be a connected, almost simple linear algebraic group defined over a finite field $\mathbb{F}_q$. Let $g, g'\in G(\mathbb{F}_q)$ be conjugates by an element of $\mathrm{GL}_n(\overline{\mathbb{...
- 20.2k
0
votes
1
answer
79
views
Matrix sparsity pattern from Boolean condition
I would like to determine the sparsity pattern of a matrix $A_m$, where the entry $(k,l)$ of $A_m$ is non-zero if the condition $\operatorname{test}(k,l,m)$ is true:
$\operatorname{test}(k,l,m)$ = &...
0
votes
0
answers
134
views
Bound on the number of connected components of a linear algebraic group $G<\mathrm{SL}_n$?
Let $G<\mathrm{SL}_n$ be a linear algebraic group defined over a field. Is there a bound on the number of connected components of $G$ in terms of $n$ alone?
(The bound will evidently not be any ...
- 20.2k
1
vote
0
answers
154
views
Some kind of product of two 2d tensors to create a 3d tensor?
I recently need to apply the following concept of product of two 2d tensors to create a 3d tensor (tensors understood as generalized arrays):
given two 2d tensors $A_{m\times n}$ and $B_{n\times p}$, ...
- 461
1
vote
1
answer
141
views
Minimal number of linearly dependent rank-1 projectors
What is the minimal number of linearly dependent rank-1 projectors $\vec v \vec v^t$ in dimension n, under the condition that every set of n column vectors $\vec v$ is linearly independent.
PS: the ...
- 1,207
0
votes
0
answers
55
views
Relationship of optimal solutions between the total function and the sub function
This is an unconstrained convex optimization problem. Let $\mathcal{N}=\left\{1,\ldots,n\right\}$, $2\leq n<\infty$. Suppose there are many strongly convex functions $f_i(x)$, where $x\in\mathbb{R}^...
- 1
0
votes
0
answers
85
views
Show that $\max_{P_X : X\in (0,1) } \left| \frac{\mathbb{E} [ f'(X) ]}{ \mathbb{E} [ f(X) ] } \right|$ is maximized by at most two mass points
Let $f$ be some given well-behaved function. Consider the following optimization problem overall probability distribution on $[0,1]$
\begin{align}
\max_{P_X : X\in [0,1] } \left| \frac{\mathbb{E} [ ...
- 671
3
votes
0
answers
452
views
Dimension of a subspace of $n\times n$ real symmetric matrices
Let $n\in \mathbb N.$ Let $W$ be a non-trivial subspace of $n\times n$ symmetric matrices such that for every $x\in \mathbb R^n\setminus \{0\}$ there exists $a_x\in \mathbb R^n\setminus \{0\}$ such ...
- 49
1
vote
1
answer
69
views
$A^*$ algorithm to find shortest path when weights in my graph are the inverse of distance
Given a graph G=(V,E) where the weights on my edges are inverse of Euclidian distance between nodes, I want to know if I can use A* algorithm to find the shortest path. How I need to modify the ...
- 111
1
vote
1
answer
176
views
Symmetric linear least-squares solution ${\bf X} {\bf A} = {\bf B}$
Given the wide matrices ${\bf A} \in {\Bbb R}^{n \times m}$ and ${\bf B} \in {\Bbb R}^{p \times m} $, where $m > n > p$, form an overdetermined linear system in ${\bf X} \in {\Bbb R}^{p \times n}...
- 11
3
votes
1
answer
342
views
Inverting a matrix using the Matrix logarithm
This is probably going to lead nowhere, but maybe it be possible to use the matrix logarithm to invert matrices?
For positive definite matrices, we have that the logarithm exists and
$$
\log(A^{-1})= -...
- 385
0
votes
1
answer
184
views
Spectrum of a product of a symmetric positive definite matrix and a positive definite operator
Let $\mathbf H$ be an infinite dimensional Hilbert space.
I want to find an example of a $2\times 2$ real symmetric positive definite matrix $M$ and a positive definite bounded operator $A : \mathbf H ...
- 63
8
votes
1
answer
879
views
Is there a conceptual reason why every square complex matrix is similar to a complex-symmetric matrix?
The question is maybe a bit vague, but like the title says: Every square complex-matrix $M$ is equal to $P S P^{-1}$ where $S = S^T$. The proof begins by taking the Jordan Normal Form of $M$, and then ...
- 4,943
1
vote
1
answer
119
views
Optimization on non-convex set
Let $\Omega$ be an open bounded subset of $\mathbb{R}^2$ and $f\in L^2(\Omega)$ be a given function. Consider the optimization problem
$$\mathrm{min} \int_\Omega u(x) f(x) \,dx\,,$$
where a minimum is ...
- 11
14
votes
4
answers
2k
views
Measuring the "distance" of a matrix from a diagonal matrix
Let $A$ be a $N \times N$ symmetric positive semi-definite matrix with $N \geq 2$. Let $D$ be a diagonal matrix of dimension $N$. I would like to measure how much $A$ "is far" from $D$, i.e. ...
- 251
0
votes
0
answers
114
views
Expectation of the operator norm of projection of a random permutation matrix
Assuming I have a fixed dimension $p$ subspace of $\mathbb{R}^d$ orthogonal to $1^d$ and $VV^\top$ with $V \in \mathbb{R}^{d \times p}$ is the orthogonal projection to the subspace.
What bound can I ...
- 23
5
votes
0
answers
180
views
Is the matrix multiplication exponent $\omega$ independent from the choice of the base field
The matrix multiplication exponent, usually denoted by $\omega_{F}$, is the smallest real number for which any two $n\times n$ matrices over a field $F$ can be multiplied together using ${\...
- 151
0
votes
0
answers
69
views
Spectrum of Moore-Penrose pseudo-inverse multiplied by a constant
Consider a random rectangular matrix $X\in\mathbb{R}^{N\times P}$ where each entry is drawn from iid distribution with mean $m$ and variance $s^2$, and denote $X^+$ the Moore-Penrose pseudo-inverse.
...
- 373
7
votes
1
answer
792
views
Remarkable recursions for the A204262
Let $a(n)$ be A204262 i.e. permanent of the matrix $n\times n$ with elements $\min(i,j)$.
Let
$$
f_{n,\ell}(x)=g_{n,\ell}(x)+f_{n,\ell-1}(\ell)-g_{n,\ell}(\ell), \\
g_{n,\ell}(x)=\int (n-\ell)^2 f_{n-...
- 4,928
8
votes
2
answers
519
views
Orthogonal basis of ${\bf Sym}_n(\mathbb R)$, made of orthogonal matrices
My question is motivated by this one, but within real matrices instead of complex ones.
${\bf Sym}_n(\mathbb R)$ is a vector space of dimension $N=\frac{n(n+1)}2$. Equipped with the scalar product $\...
- 52.3k
3
votes
1
answer
161
views
Elastic network model Hessian rigid body motion 0 eigenvalues
Based on a method that apparently seems to be widely used in computational chemistry (cf https://en.wikipedia.org/wiki/Anisotropic_Network_Model)
Trying to build a very simple model with 3 atoms ...
- 181
2
votes
1
answer
98
views
One question about nega-cyclic Hadamard matrices
Let $n$ be a multiple of $4$, is there any $n \times n$ negacyclic Hadamard matrix? If yes - how to construct it? If no - why?
Here an $n \times n$ nega-cyclic matrix is a square matrix of the form:
\...
- 696
5
votes
1
answer
141
views
On the half-skew-centrosymmetric Hadamard matrices
Definition 1: A Hadamard matrix is an $n\times n$ matrix $H$ whose entries are either $1$ or $-1$ and whose rows are mutually orthogonal.
Definition 2: A matrix $A$ is half-skew-centrosymmetric if ...
- 696
0
votes
0
answers
79
views
Approximate solution problem of rank-one modification matrix secular equation
In Golub's paper , page 327,the eigenvalues of a rank-one modification of a $n\times n$ symmetric matrix can be computed by findng the zeros of the secular equation
\begin{equation*}
w(\lambda_j)=...
- 63
1
vote
0
answers
122
views
Hard instances for this graph isomorphism algorithm based on powers of weighted adjacency matrices?
In short, I found an algorithm for GI and the only hard instances
I found so far are non-isomorphic strongly regular graphs with
large automorphism groups.
Q1 What are hard instances for the ...
- 25.4k
6
votes
1
answer
288
views
The combinatorics of the Nullstellensatz for the variety of nilpotent matrices
Let $H_n$ denote the set of $n \times n$ nilpotent matrices with complex entries. The set $H_n$ may be regarded as an algebraic variety. Indeed, consider the polynomial ring $\mathbb{C}[A_{i,j} : 1 \...
- 484
5
votes
1
answer
136
views
Estimating a symmetric positive-definite matrix from list of matrix vector products
I have a symmetric positive definite matrix (hessian) $H$ which is unknown and expensive to compute explicitly (circa 30*30)
Indirectly in my code I have a growing list of pairs of unit vectors $u_i$, ...
- 71
8
votes
1
answer
323
views
On a matrix inequality
$\newcommand{\R}{\mathbb R}\newcommand{\tr}{\operatorname{tr}}$It follows from Proposition 7 and this recent answer that, for any positive-definite $n\times n$ symmetric real matrices $A$ and $B$,
$$\...
- 128k