All Questions
6,289 questions
2
votes
0
answers
138
views
Is $L^2([a,b]; L^2(S^2))$ the same as $L^2([a,b] \times S^2)$?
The space $L^2([a,b];L^2(S^2))$ is a Banach space with respect to the norm
$$\left\Vert f \right\Vert_1^2 = \int_{a}^b \left\Vert f(r) \right\Vert_{L^2(S^2)}^2 dr$$
The space $L^2([a,b]\times S^2)$ ...
- 969
1
vote
0
answers
95
views
Vandermonde-type factorization of moment matrix?
Consider $n,d \in \mathbb{N}_{>0}$, there are many functions $y:\mathbb{N}^{n} \to \mathbb{R}$. Now for simplicity, we denote $y(\alpha)$ to be $y_{\alpha}$. Let $|\alpha| = \sum_{i=1}^{n}\alpha_{i}...
- 275
0
votes
0
answers
68
views
Meaning of $\langle M,\,M^{-1}\rangle$
For an $n\times n$ complex matrix $M$, is there a name for the expression $\langle M,\,M^{-1}\rangle$, where the inner product is the Frobenius one, $\langle A,\,B\rangle=\text{tr}(A^*B)$? Is there a ...
- 479
3
votes
0
answers
106
views
Bijectivity of a linear map between symmetric polynomials of even degree
Let $\mathfrak S_n$ be the symmetric group of permutations of $n$
letters and let $S = \sum_{\sigma\in\mathfrak S_n} \sigma$ be the
symmetrization operator.
Let $\Lambda_n^r$ be the vector space of ...
- 5,822
0
votes
0
answers
55
views
Johnson-Lindenstrauss type result for matrix factorization
The type of result I want is: given matrix $A\in \mathbb{R}^{m\times n}$ and error tolerance $\epsilon$, what is a lower bound on $k$ such that $\|A - UV\|_{??}\le \epsilon$, where $U \in\mathbb{R}^{m\...
2
votes
0
answers
60
views
Basis vectors using anti-commuting operators?
Let $V$ be a finite-dimensional inner product space. Suppose $A_{1},...,A_{N}$ are anti-commuting operators, meaning that these are linear operators on $V$ that satisfy:
$$A_{i}A_{j}+A_{j}A_{i} = A_{i}...
- 1,305
1
vote
2
answers
66
views
Distribution of the constraint matrix conditioned on the solution of the linear system
Suppose that A is a random matrix in $R^{n\times n}$, with each component independently and identically distributed (iid) according to $\mathcal{N}(0,1)$. Additionally, b is a random vector in $R^n$, ...
- 33
5
votes
2
answers
420
views
Maximum determinant of binary matrices with special properties
Let $A$ be an $n$-by-$n$ binary matrix (all elements are in $\{0,1\}$). It is known that the determinant of $A$ is bounded by $O(n^{n/2} / 2^n)$. I am looking for tighter upper bounds for matrices ...
- 3,549
0
votes
0
answers
43
views
Given two rectangular matrices and they yield the same results when they are multiplied by their own transposes. What can we say about them?
Suppose we have $MM^T = NN^T$, where $M$ and $N$ are both $n$ by $d$ matrices. Assume that $n$ is (much) larger than $d$, are there anything we could conclude about $M$ and $N$, aside from that $N$ ...
- 9
5
votes
1
answer
349
views
Commutator subgroup of $\mathrm{GL}_n(R)$ when $R$ is a PID with unity
Hua and Reiner in their paper titled "Automorphisms of the unimodular group" have established what will be the commutator subgroup of $\mathrm{GL}_n(\mathbb{Z})$, $\forall n\geq 0$. I have ...
- 51
1
vote
0
answers
37
views
When does an optimal input sequence for a discrete-time system exist?
Suppose an LTI discrete-time system is given by the equations
$$
x_{k+1} = Ax_k + Bu_k,\\
y_{k} = Cx_k + Du_k
$$
with $x_k\in\mathbb{R}^{m}$, $y_k\in\mathbb{R}^{n}$ and $u_k\in\mathbb{R}^{p}$ and $\...
1
vote
0
answers
73
views
What is the closed form of a polyhedral cone's dual cone?
A polyhedral cone can be defined as
$$
\mathcal{K} = \{x~|~Ax\preceq 0\},
$$
where $A \in \mathbb{R}^{m \times n}$, $x\in \mathbb{R}^n$ and $\preceq$ denotes component-wise less than and equal to.
The ...
- 71
0
votes
0
answers
64
views
When is a symmetric block Toeplitz matrix invertible?
Let
$$
Q =
\begin{bmatrix}
Q_0 & Q_1 & Q_2 & \cdots\\
Q_{-1} & Q_{0} & Q_1 & \cdots\\
Q_{-2} & Q_{-1} & Q_0 & \cdots\\
\vdots & \vdots & \vdots & \ddots
...
1
vote
0
answers
29
views
Can all real positive semidefinite Hankel matrices be decomposed into sum of rank 1 real positive semidefinite Hankel matrices?
Denote the set of real positive semidefinite $d\times d$ Hankel matrices as $\mathcal{S}$. Can we always decompose one $S\in \mathcal{S}$ into sum of rank $1$ $S_i\in \mathcal{S}$, i.e., $S=\sum_i S_i$...
- 11
0
votes
0
answers
28
views
Find a conditional for factorizing the sum of a set of gaussian integer-valued matrices
In my research project, we're exploring the decomposition of Gaussian integer-valued square matrices as a cross-product of other Gaussian integer matrices (GIM) with the same dimension. One of the ...
- 11
13
votes
1
answer
580
views
Identifying two definitions of orientation on a vector space
Let $V$ be an $n$-dimensional real vector space. Here are two definitions of an orientation on $V$:
A generator of the $1$-dimensional vector space $\wedge^n V$, up to multiplication by positive ...
- 133
0
votes
0
answers
87
views
Relation between nullspace and row-equivalence of matrices over $\mathbb{Z}$ and $\frac{\mathbb{Z}}{n \mathbb{Z}}$?
Two matrices $D$ and $E$ over a field have the same nullspace if only if they are row-equivalent. Is the same true if those matrices are over the ring of integers ($\mathbb{Z}$) or integers mod a ...
- 219
0
votes
0
answers
75
views
Orbits/affine spaces in GAP
Another GAP-related question.
I need to compute the orbits of a lot (probably, hundreds of thousands) groups acting on $\mathbb{F}_2$-vectors spaces of dimension 23 or 22. The groups range from (...
- 4,959
0
votes
0
answers
43
views
Inertia indices in GAP
Not sure that this is the right place, but I could not find a GAP specific forum.
Does anyone know if there is a built-in function in GAP to find the inertia indices of a symmetric matrix, say, over ...
- 4,959
2
votes
1
answer
94
views
Testing for equal characteristic polynomials using a single determinant calculation
Let $A_1,A_2$ be $n\times n$ symmetric matrices over $\{0,1\}$, and let $p_1, p_2$ be their respective characteristic polynomials over the rationals.
If $p_1 \ne p_2$, then there is some positive ...
- 37.7k
0
votes
0
answers
121
views
Representation of anti-commuting matrices in $\mathbb{C}^{2}$
This is a cross posting updated question from MSE. I have not got any answers there yet and I really want to understand this problem.
The basic question is the following. Let $V$ be a finite-...
- 1,305
1
vote
1
answer
48
views
Iteration matrix representation with complex conjugate operator
I am studying the convergence of a particular class of radial power flows, whose goal is to obtain the voltage solution for a given electric grid, i.e., a complex vector $\mathbf{V}$ that gives the ...
0
votes
1
answer
103
views
Convex sets via fixed point equations
I have an equation of the general form
$$ X = S \cup T X $$
where $S \subset \mathbb R^n$ is a convex polytope (given by its bounding hyperplanes), $T\colon \mathbb R^n \to \mathbb R^n$ is a linear ...
- 857
0
votes
0
answers
71
views
Reference request for equivalent Lipschitz smoothness conditions
For an open set $Z\subseteq\mathbb{R}^n$, let $f: Z\mapsto \mathbb{R}$ be a continuously differentiable function on $Z$, and let $L>0$ be fixed. Also, suppose that (a) $f$ is nonconvex and (b) $f$ ...
3
votes
2
answers
392
views
Monotonicity of matrix conjugation
Let $A$ and $B$ be positive-definite matrices such that $A \le B.$
By matrix monotonicity of the root, this also implies that $A^{\alpha} \le B^{\alpha}$ for $\alpha \in [0,1].$
I am now curious under ...
0
votes
1
answer
170
views
Isn't every algebraic operad equipped with a trivial weight?
In Loday–Vallette "Algebraic Operads" they state the following result (Theorem 6.6.2, Operadic twisting morphism fundamental theorem):
Let $P$ be a connected weight graded differential ...
- 215
3
votes
1
answer
547
views
Why is this polynomial factorizable? [closed]
I met a curious problem on factorizing a homogenerous polynomial of degree 9.
Problem: Show that the following polynomial can be divided by $(a_1+a_2+a_3)$:
\begin{align}
&\quad\left|
\begin{array}...
- 357
3
votes
0
answers
58
views
About a circular variant of Vandermonde matrix
Given an arbitrary $(x_1, \dots, x_n) \in [0, 1]^n$, is there any name/known results for the following $n \times n$ matrix (which is constructed by iterating $(x_1 \to \dots \to x_n \to x_1 \to \dots)$...
- 33
1
vote
2
answers
152
views
Property for bounding matrix exponential
Wikipedia states in the exponential map section about the exponential of a matrix that for any matrices $X$, $Y$ it holds that $\|e^{X+Y}-e^{X}\| \leq \|Y\|e^{\|X\|} e^{\|Y\|}$ where $\|\cdot\|$ ...
- 21
11
votes
2
answers
550
views
Let $a_1, \dots, a_n$ be a finite set of positive reals. Is there a $\mathbb Q$-basis of $\mathbb R$ where each $a_i$ has nonnegative coordinates?
Let $a_1, \dots, a_n$ be a finite set of positive reals. Is there a $\mathbb Q$-basis of $\mathbb R$ where each $a_i$ has nonnegative coordinates?
Playing around with the case $n = 2$, I’m pretty sure ...
- 63.9k
0
votes
1
answer
222
views
Low rank matrix in a subspace of matrices
Let $V\subset M_{m,n}(\mathbb{C})$ be a $n$-dimensional subspace ($m\geq 2$).
Is there $X\in V$ such that $1\leq \operatorname{rank} X\leq m-1$?
- 13
1
vote
0
answers
130
views
A basis of the weight space in the semi-invariant ring corresponding to the weight $\langle(2,3,2),\cdot\rangle$
I'm trying to understand Example 10.11.1 on page 225 of the book "An introduction to quiver representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below)
I want to ...
- 839
2
votes
0
answers
101
views
On the irreducible submodules of adjoint representations $\text{ad}^{0}$
Let $k$ be a finite field of characteristic $p$. Let $H$ be a subgroup of $\rm{GL}_{n}(k)$ of order prime to $p$ where $n\geq2$. Assume that the representation $H\hookrightarrow \rm{GL}_{n}(k)$ is ...
- 667
2
votes
1
answer
99
views
Stabilizing conjugacy classes of integer matrices
$\DeclareMathOperator{\Conj}{Conj} \DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\id}{id} \newcommand\Z{\mathbb{Z}}$
For an $n \times n$ integer matrix $A \in \GL_n(\Z)$, let $\Conj(A)$
be the ...
- 23
0
votes
0
answers
84
views
some problem about the discrete of the first derivative operator
I am reading a paper
(Parameter Choice Strategies for Multipenalty Regularization Massimo Fornasier, Valeriya Naumova, and Sergei V. Pereverzyev SIAM Journal on Numerical Analysis 2014 52:4, 1770-1794)...
- 33
0
votes
0
answers
67
views
Concentration of bilinear forms
This is a bit vague so I'll begin by indicating the motivation. I am looking for ways to [do something interesting or useful] with the self-attention in transformer models. Ultimately the self-...
- 6,962
1
vote
1
answer
205
views
Van der Waerden conjecture and Alexandrov-Fenchel inequality
The Van der Waerden conjecture is a lower estimate of the permanent of a doubly stochastic matrix. In this article in Wikipedia it is stated that Egorychev's proof uses the Alexandrov-Fenchel ...
- 21.8k
3
votes
1
answer
153
views
Solving a recursion for polynomials defined by a matrix product
Define the polynomial $p_n(X) \in \mathbb{Z}[X_1,...,X_n]$ as the top left entry in $A^n$ for the $(d \times d)$ matrix
\begin{align*}
& A = \left(\begin{matrix}
X_1 & \dots & \...
- 1,295
3
votes
0
answers
83
views
A stochastic matrix $B = \lambda(\lambda I - A)^{-1}$ such that $B-B^2$ has a non-negative diagonal
I apologize if this is too elementary a question, but I have not been able to make much progress.
Consider a real matrix $A$ with $A_{ij} >0$ for $i \ne j$ and $\sum_{j} A_{ij} = 0$ for each $j$. ...
- 251
0
votes
2
answers
62
views
Nondegeneracy of dominant singular value and positivity of dominant singular vector of connected nonnegative matrix
Call a (not necessarily square) nonnegative matrix $M$ connected if there do not exist permutation matrices $P$ and $Q$ such that $PMQ=\begin{pmatrix}A&0\\0&B\end{pmatrix}$ for some $A$ and $B$...
- 674
0
votes
0
answers
183
views
Degree 6 Galois extension over $\mathbb{Q} $
Let L be the splitting field of $ x^3- 2$ over $ \mathbb{Q}$. Then $ G=\operatorname{Gal}(L/K) \cong S_3$. Let $\sigma\in G$ such that the fixed field of $ \sigma$ is $\mathbb{Q}(2^{1/3})$. Let $x,y\...
- 923
2
votes
2
answers
235
views
$L^p$ domination of mixed partial derivatives of the 3rd order by the unmixed ones?
Is it true that for each real $p>1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^3$ one has
$$\|D_1D_2D_3u\|_p\le C_p(\|D_1^3u\|_p+\|...
- 128k
5
votes
0
answers
107
views
Generalized Puiseux series for diagonal reflections of the curves $y = \frac{x}{(1-ax)(1-bx)^m}$
Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ ...
- 10.5k
2
votes
2
answers
197
views
$L^p$ domination of mixed partial derivatives by the unmixed ones?
Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has
$$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
- 128k
2
votes
0
answers
81
views
Given a low-rank symmetric positive semidefinite matrix and a basis of its nullspace, is there a fast way to get the nonzero eigenvalues?
I have a (possibly dense) $k\times k$ real matrix $L = AA^T + B^T B$, a type of combinatorial Laplacian (self-adjoint, symmetric, positive semidefinite) of rank $(k-n)$ and possibly repeated nonzero ...
- 21
2
votes
1
answer
278
views
Continuity of eigenvector of zero eigenvalue
Wonder whether anyone has an idea on showing the following or to point out that it is not true:
Let $A(t) \in \Re^{n \times n}$ be differentiable over an interval $I$, and it has a zero eigenvalue for ...
- 69
4
votes
2
answers
180
views
What does the matrix-valued solution of $X = A X A^T + \operatorname{Id}$ look like?
Let $A \in \mathbb{R}^{n \times n}$ be an invertible contraction, i.e. all singular values are in $(0,1)$. By reformulating the equation
\begin{align*}
& X = A X A^T + \operatorname{Id} \tag{1}
\...
- 1,295
2
votes
0
answers
60
views
Perron-Frobenius theory for operators on matrices
Let $A$ be a Hermitian linear operator on the space of $n\times n$ complex matrices. Let's suppose $A$ is "non-negative" in the sense that it preserves the cone of non-negative definite (...
- 53
2
votes
1
answer
87
views
Symmetric and anti-symmetric matrices and maximal eigenvalues
Suppose we start with a symmetric $n \times n$ matrix $A$, the elements of which are either $1$ or $0$. All the diagonal elements of this matrix are set to be $0$. So, $\lambda_{\text{max}}=\sup \...
- 1,755
9
votes
3
answers
350
views
$G$-module structure of the relation module for a presentation of a finite group $G$
Let $F_n$ be a free group of rank $n\ge 2$, and $F_n\rightarrow G$ a surjection with $G$ finite. Let $R$ be the kernel. From this, we get an action of $G$ on the abelianization $R/R'$ (a free abelian ...
- 7,527