All Questions
6,028 questions
3
votes
0
answers
59
views
Maximizing a Gaussian quadratic form
Let $u$ denote a fixed unit vector in $\mathbb{R}^n$ and $g$ a standard Gaussian vector (in $\mathbb{R}^n$).
Consider the map
$$
f_n(X) = \mathbb{E} \langle (X^{-1} + gg^T)^{-1} u, u\rangle,
$$
...
- 420
0
votes
0
answers
66
views
Taking trace of a tensor product of matrix-valued smooth functions on the thin diagonal
Let $V$ be a finite dimensional real / complex vector space and consider the space $L(V,V)$ of linear operators on $V$.
Fix $n \in \mathbb{N}$ and let $\mathcal{M}$ be the real / complex vector space ...
- 3,477
2
votes
1
answer
210
views
Maximum number of ones in a full rank matrix with a restriction
Consider $n \times n$ binary matrices. I am interested in the largest number of ones possible in an $n \times n$ binary matrix with full rank over the field of integers mod 2 with the following ...
- 3,377
1
vote
1
answer
91
views
Positive definite kernels on compact interval $[0,1]$
From How to prove that a kernel is positive definite? I learned that a function $f:[0,\infty)\to\mathbb{R}$ induces a positive definite kernel $K:\mathbb{R}^2\to\mathbb{R}$, $K(x,y)=f((x-y)^2)$ if $f$ ...
- 216
2
votes
1
answer
312
views
Question on a vector inequality
Is it true that
$$
\min\left( \begin{aligned}
&\|\mathbf{u}\| + \|\mathbf{v}\| - \|\mathbf{u} + \mathbf{v}\|, \\
&\|\mathbf{u}\| + \|\mathbf{w}\| - \|\mathbf{u} + \mathbf{w}\|, \\
&\|\...
- 171
2
votes
0
answers
89
views
The linear independence and linear elimination of non-crossing matching polynomials
Consider the polynomial set:
$$
f_{ij} = (t_i - t_j)x_i x_j + x_i - x_j, \quad (1 \leq j < i \leq 2n)
$$
where $ t_1, t_2, \dots, t_{2n} $ are pairwise distinct.
Let's look at the non-crossing ...
- 21
24
votes
3
answers
866
views
Mark some vectors in $\mathbb{R}^n$ in a way that every orthonormal basis has an odd number of marked vectors
Let $n$ be a natural number. Is there a set $S$ of vectors of norm $1$ in $\mathbb{R}^n$ such that every orthonormal basis of $\mathbb{R}^n$ contains an odd number of vectors from $S$?
If $n$ is odd, ...
- 679
8
votes
1
answer
361
views
Invertible matrix with group ring coefficient
Before asking the question I do need
some notations.
$G$ a (torsion-free) group, $\mathbb{Z}^{´}=\mathbb{Z}[\frac{1}{2}]$
$R:= \mathbb{Z}[G]$, $R^{´}=\mathbb{Z}^{´}[G]$ group rings.
$Mat_{n}(R)$ the ...
- 223
7
votes
3
answers
958
views
Vector of integers such that almost all dot products are positive
Let $x_1<x_2<\cdots<x_n$ be $n$ real numbers such that $\sum\limits_{j=1}^n x_j\ne0$. Do there always exist $n$ integers $a_1,a_2,\ldots,a_n$ such that
$$
\sum_{j=1}^n a_j\cdot x_j <0
\...
- 3,153
20
votes
1
answer
557
views
Almost orthogonal maps $f:\omega \to \{-1,1\}$
Let $\omega$ denote the set of non-negative integers. For sets $A,B$, let $B^A$ denote the set of maps $f:A\to B$. For $f,g\in\{-1,1\}^\omega$ we say that $f,g$ are almost orthogonal if there is $C_0\...
- 48.9k
3
votes
0
answers
181
views
Levelled trees and the homotopy transfer theorem
In section 10.3.12 of Loday-Vallette's book "Algebraic operads", given a $P_\infty$-algebra $(A,d,\alpha)$ the Homotopy Transfer Theorem applied to $H_*(A,d)$ is studied. There, because the ...
- 215
1
vote
0
answers
93
views
Similarity of non-standard matrices
I am researching numerical methods for PDEs. I particular, I am looking at methods for the linear hyperbolic PDE
$$
u_t+au_x=0.
$$
This is a common approach, because successful methods for this model ...
- 111
6
votes
2
answers
492
views
Does this polynomial have a real zero less than or equal to $1/2$?
Is the smallest root $x$ of
$$
10x^{3}-30x^{2}+\left(30-2\sum_{1\le i<j\le6}\cos^{2}\alpha_{ij}\right)x\\
+2\sum_{1\le i<j\le6}\cos^{2}\alpha_{ij}-\sum_{1\le i<j<k\le6}\cos\alpha_{ij}\cos\...
user538574
3
votes
2
answers
303
views
Asymptotics of A000613
The general linear group $GL_n(\mathbb{F}_2)$ acts on the powerset $2^{{\mathbb{F}_2}^n \setminus \{0\}}$ by multiplication: $A \cdot S := \{Ax \in {\mathbb{F}_2}^n : \, x \in S\}$, for an invertible ...
- 331
1
vote
0
answers
40
views
Asymptotic unitary invariance of rank-one spiked Gaussian matrix
I'm working on some Random Matrix Theory related stuff for my thesis, and i've come across the following problem:
Consider a (normalized) spiked Wigner matrix $\mathbf{A}$
$$ \mathbf{A} = \frac{\beta}{...
- 11
0
votes
1
answer
140
views
Finding positive vectors of a special LGS
Let the following $4 \times 4$ LGS be given for which all coefficients
$a_1, a_2, a_3, a_{11}, a_{12}, ..., a_{33}$ are $>0$:
$a_1 + a_{11} \; x_1 + a_{12} \; x_2 + a_{13} \; x_3 + 0 \; x_4 = (a_{...
15
votes
1
answer
649
views
On minimal eigenvalue
Is it true that $\min\left(\lambda_{\min}(M_{12}),\lambda_{\min}(M_{13}),\lambda_{\min}(M_{23})\right) \le \frac{7}{20}$ where $M_{ij}$ is the matrix obtained by selecting the entries at the ...
- 178
2
votes
0
answers
67
views
Preserving invertibility with adding rows
Suppose I have two $m\times n$ matrices $A$ and $B$ such that an $m\times m$ submatrix of $A$ is invertible if and only if the corresponding $m \times m$ submatrix of $B$ is. Now let's say I append a ...
- 21
2
votes
1
answer
315
views
Are surjective homogeneous maps open at zero?
I'm asking this question as a follow-up inspired by this one: An open mapping theorem for homogeneous functions?
I'm actually wondering whether there exists an homogeneous map $f:\mathbb R^n\to\mathbb ...
- 311
7
votes
4
answers
559
views
Reference request: "Higher order eigentuples" as generalized eigenvectors?
I stumbled upon a cute generalization of the eigenvalue problem and would like to know if anybody has seen something like this and can provide references.
The eigenvalue problem for a square matrix $M$...
- 12.7k
5
votes
1
answer
303
views
Efficiently computing $\prod_{i=1}^{n} A_i$
Let $k$ be a nonnegative integer, how to compute $\prod\limits_{i=1}^{n} A_i$ quickly and accurately, where $$A_i=\begin{bmatrix}
0 & 1\\
i^k & 1
\end{bmatrix}?$$
I know if $k=0$, we can use ...
- 696
10
votes
3
answers
455
views
When does $\det(\frac{A+A^T}{2})=\det(A)$ for positive-definite $\frac{A+A^T}{2}$?
Setup: Let $A$ be a real square matrix and assume its symmetric part $\frac{A+A^T}{2}$ is positive-definite. The inequality
$$
\det\left(\frac{A+A^T}{2}\right) \leq \lvert\det(A)\rvert
$$
is known as ...
- 145
1
vote
0
answers
76
views
What is the operator norm of the sedenions and beyond?
Suppose that $K$ is a field. Then for all $n$, define a bilinear operation $*$ (or $*_{n,K}$ in case there may be ambiguity) on $K^{2^n}$ along with a conjugation operation $^*$ on $K^{2^n}$ by ...
0
votes
1
answer
91
views
Finite projective geometry and the Krasner hyperfield
The Krasner hyperfield is an algebraic structure of two operations on $K=\{0,1\}$ called $+\colon K\times K\to \mathcal{P}(K)$ and $\cdot\colon K\times K\to K$ with
$0+0=0$
$0+1=1+0=1$
$1+1=\{0,1\}$
...
- 10.4k
1
vote
0
answers
204
views
The wedge product of two positive forms is positive
I have previously posted this question on MSE, but still didn't solve it.
Definition. A real $(p, p)$-form $\psi$ on a complex manifold $M^{n}$ is said to be (semi-) positive, if for any $x \in M$, ...
- 195
21
votes
0
answers
520
views
Is the exponent of $2$ in the Pythagorean theorem the "same $2$" as $[\mathbb{C} : \mathbb{R}]$?
I posted this question in Math StackExchange a couple years ago; due to the recent surge in interest, and following the feedback of several users, I've decided to cross-post it here. I apologize for ...
- 1,206
0
votes
1
answer
102
views
Minimally change matrix with determinant 0
In the following matrix equation, all coefficients $a_{ij}>0$ and all $a_i>0$ and the column sums in the matrix $A$ are all 0
(e.g. $-a_{11}+a_{21}+a_{31}=0$, etc.).
This means that
the ...
2
votes
0
answers
331
views
What is the spectrum of this differential operator?
My self-adjount differential operator $L$ is defined by $$L f(x) \equiv u(x) \frac{\partial^2}{\partial x^2} \left( u(x) f(x) \right)$$ where $u(x)$ is a known but arbitrary smooth function that ...
- 151
3
votes
1
answer
153
views
Number of points covered by $2n$ hyperplanes in $\mathbf{F}_p^n$
For a prime $p$, fix two bases $U=\{v_1,\dots,v_n\}$ and $W=\{w_1,\dots,w_n\}$ of the vector space $V=\mathbf{F}_p^n$. We may assume $U$ is the standard basis without loss of generality.
For $s_1,\...
- 281
89
votes
1
answer
21k
views
Is the largest root of a random polynomial more likely to be real than complex?
This question might be hard because it got $35$ upvotes in MSE and also had a $200$ points bounty by Jyrki Lahtonen but it was unanswered. So I am posting it in MO.
The number of real roots of a ...
- 5,470
0
votes
1
answer
114
views
Geometric interpretation of a Grammian-like function
Let $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ and consider the following function $f : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$:
$$
f(\mathbf{v},\mathbf{w}) = \|\mathbf{v}\|\|\mathbf{w}...
- 6,351
2
votes
1
answer
158
views
The relationship between a matrix and its coefficient matrix decomposed in Pauli matrix
For a dimension-$4$ Hermitian matrix $A$, denote pauli matrices $\{I,X,Y,Z\}$ as $\{\sigma_0,\sigma_1,\sigma_2,\sigma_3\}$ respectively. The pauli matrices form a basis of the matrix space if we take ...
- 91
1
vote
0
answers
100
views
PageRank in directed graphs: equivalence of iterative and eigenvalue methods
Given a directed graph $ G $ with $ n $ nodes, we can represent this graph using an adjacency matrix $ A $. The stochastic matrix $ S $ can be derived from the adjacency matrix using the following ...
- 4,058
1
vote
1
answer
142
views
Operator norm of some type of discrete Fourier matrix
Let $N$ be a natural number and let $w$ be a complex number.
We define the $N\times N$ matrix $C_w=(a_{k,l})_{k,l=1}^N$ as follows,
$$
a_{k,l}=\begin{cases}1 & l=k+1\\
w &...
- 4,058
2
votes
1
answer
172
views
Diagonalize almost symmetric tridiagonal matrix
I begin with an $n \times n$ real symmetric tridiagonal matrix. However, I replace the non-zero elements in the first and last rows with zeros, so it is no longer symmetric
$$M = \begin{bmatrix} 0 &...
- 151
0
votes
0
answers
60
views
The generalized Laplace expansion for tensor
I'm reading this paper https://arxiv.org/abs/1308.3860.
In the Appendix (page 22), the author uses a generalized Laplace expansion for the determinant tensor, as shown in the picture1.
But I only ...
- 1
2
votes
1
answer
474
views
Polynomial $f(x)$ has positive coefficients and only real roots. How many polynomials formed from terms of $f(x)$ also have only real roots?
Let
$$f(x)=a_n \ x^n+a_{n-1} \ x^{n-1}+\cdots+a_1 \ x+a_0$$
be a $n$-th degree polynomial with positive coefficients such that all of its roots are real. Choose any number terms from this expression ($...
1
vote
1
answer
145
views
Prove or disprove that the matrix equation of the form $AX+XA^{-T}=0$ has a nonsingular anti-symmetric solution $X$
I’m trying to prove that for $A=J_n(i)$, that is, the Jordan block matrix corresponding to the eigenvalue $i$ of size $n$, where $n$ is even, the matrix equation $AX+XA^{-T}=0$ has a nonsingular anti-...
- 11
4
votes
1
answer
190
views
Is the transpose of an infinite Hadamard matrix also Hadamard?
Let $\omega$ be the set of non-negative integers. If $f,g:\omega\to\{-1,1\}$ are maps, then we say $f,g$ are almost orthogonal if there is a positive integer $C_0\in \omega$ such that for all $n\in\...
- 48.9k
1
vote
1
answer
151
views
How to prove that each element of $A(A^TA)^{-1}A^Ty$ is greater than 0, if $A(i,j)>0$ and $y=[1, 1, 1, ..., 1]^T$
Let $A\in \mathbb{R}^{m\times n}$, $m>n$, $rank(A)=n$, and $\forall 1 \leq i \leq m, 1 \leq j \leq n, A(i, j)>0$, $y=[1, 1, 1, ..., 1]^T$. Let $\beta=A(A^TA)^{-1}A^Ty$, how to prove that each ...
- 27
0
votes
1
answer
127
views
Under what conditions does $x^TA^{-1}y> 0$ hold? $A$ is a symmetric positive definite matrix,$A\in \mathbb{R}^{n\times n}_+, x,y\in \mathbb{R}^{n}_+$
This is a tricky problem I encountered in my research. $A\in \mathbb{R}^{n\times n}_+, x,y\in \mathbb{R}^{n}_+$, i.e. $\forall 1\leq i \leq n, 1 \leq j\leq n, A(i, j)>0, x(i), y(i)>0$.
As known, ...
- 27
368
votes
31
answers
80k
views
Geometric interpretation of trace
This afternoon I was speaking with some graduate students in the department and we came to the following quandary;
Is there a geometric interpretation of the trace of a matrix?
This question ...
3
votes
1
answer
327
views
Derivative norm estimates
Assume $\Phi$ is some diffeomorphism of a certain manifold. Let $\Phi^{-1}$ denote the inverse map and let $(D\Phi)^{-1}$ denote the matrix inverse of $D\Phi$.
QUESTION. Does this norm estimate hold? ...
- 43.2k
14
votes
2
answers
852
views
Examples of finitely presented subgroups of $\operatorname{GL}(n,\mathbb{Z})$ with unsolvable decision problems
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Does there exist a finitely presented subgroup of $\GL(n,\mathbb{Z})$ for which it is known that the conjugacy problem is unsolvable (if yes, ...
- 143
4
votes
1
answer
103
views
When do the nonzero eigenvalues of a directed graph Laplacian have the same absolute value?
Question: Let $G$ be a strongly connected directed graph on $n$ vertices with Laplacian $L(G)$. Then $L(G)$ has one zero eigenvalue $\lambda_1=0$ and $n-1$ nonzero eigenvalues $\lambda_2,\ldots,\...
- 145
4
votes
1
answer
230
views
$\omega\times\omega$-Hadamard matrices
In the following, we define infinite Hadamard matrices.
Let $\omega$ be the set of non-negative integers. If $f,g:\omega\to\{-1,1\}$ are maps, then we say $f,g$ are approximately orthogonal if $$\...
- 48.9k
0
votes
0
answers
121
views
Closed form of coefficients of a finite field polynomial
I want to find a valid polynomial for a finite field $\mathbb{Z}_p[x]_{f(x)}$ with $d=deg(f(x))$. For this definition to hold, it can be deduced that $p$ must be prime and the polynomial $f(x)$ ...
- 47
2
votes
2
answers
227
views
Is a probabilistic implementation of unitaries invertible?
Let $\{p_j\}_j$ be a set of probabilities, $\sum_j p_j = 1$, let $\{h_j\}_j$ be a set of $n \times n$ Hermitian matrices, and define $ad_h(A) $ be the adjoint.
Define the following linear mapping
$$ E(...
- 131
8
votes
4
answers
379
views
Traceless Hermitian matrices with simultaneously vanishing Rayleigh quotients
Let $D$ be an integer greater than 1. What is the largest number $N$, such that for all sets of $N$ Hermitian $D\times D$ traceless matrices $M_i$, $i=1,\dots,N$, there exists a non-zero complex ...
- 628
0
votes
0
answers
46
views
What's the problem in using spanning Bessel sequences that are not frames to decompose vectors?
This is related to a question I recently asked on math.SE.
Consider a subset $G\equiv \{g_k\}_{k\in\mathbb{N} }\subseteq\mathcal H$ in a separable Hilbert space $\mathcal H$, and suppose $G$ spans the ...
- 342