All Questions
29 questions from the last 30 days
7
votes
1
answer
491
views
Invertibility of a matrix defined using inner product
Let $n,m \geq 1$. We fix $n$ distinct vectors $x_1, ... , x_n \in \mathbb{R}^m$. We define $A \in \mathbb{R}^{n\times n}$ as
\begin{equation}
A_{ij} = x_i^T \left(n x_j - \sum_{1 \leq k \leq n} x_k \...
- 2,306
6
votes
2
answers
509
views
Is every automorphism of a cone diagonalisable?
Let $V$ be a real, finite-dimensional vector space, and let $C\subset V$ be a closed convex cone, with nonempty interior, and such that $C\cap (-C)=\{0\} $. Let $u\in\operatorname{GL}(V) $ such that $...
- 38k
4
votes
2
answers
206
views
Normalizers of the principal congruence subgroups in $\mathrm{GL}(n,\mathbf Q)$
A question quite similar to this question. Let $n \geqslant 3$ and $m \geqslant 2$ be natural numbers and suppose that a matrix $A \in \mathrm{GL}(n,\mathbf Q)$ normalizes the principal congruence ...
- 43
7
votes
2
answers
242
views
Prove that $ n \leq d+1 $ under ordering constraints in $\mathbb{R}^d$
Let $x_1, \dotsc, x_n \in \mathbb{R}^d$ and $\theta_1, \dotsc, \theta_n \in \mathbb{R}^d$ be vectors such that for every $k \in [n]$, the following inequality holds:
$$
\langle x_k, \theta_k \rangle &...
8
votes
1
answer
358
views
Eigenvalues of a certain combinatorially defined matrix
Let $A_n$ be the matrix whose rows and columns are indexed by pairs
$(i,j)$ with $1\leq i,j\leq n$ and $i\neq j$ (so $A$ is an
$n(n-1)\times n(n-1)$ matrix), whose $((i,j),(k,l))$-entry is 0 if
$i=k$ ...
- 50.8k
3
votes
1
answer
200
views
Matrix-tree theorem for inverse matrices
Let $L$ be the Laplacian of a directed weighted graph on $n$ nodes, e.g., for $n=4$:
$$
L = \left(\begin{array}{cccc} w_{1,1}+w_{1,2}+w_{1,3}+w_{1,4} & -w_{1,2} & -w_{1,3} & -w_{1,4}\\ ...
- 20.2k
9
votes
1
answer
292
views
What are the points of the algebra of polynomial functions on an arbitrary vector space?
Let $V$ be an arbitrary vector space over some field $\mathbb{K}$ (UPD: of characteristic 0), $V^*=\mathrm{Hom}(V,\mathbb{K})$ its linear dual. Let $\mathrm{Sym}_\mathbb{K}(V^*)$ be the free ...
- 480
4
votes
1
answer
172
views
Maximizing trace subject to two equality constraints
I am looking at the following optimization problem
$$\begin{align}
\underset{{\bf X}}{\text{maximize}} \qquad&\mathrm{tr}({\bf AX})\\
\text{subject to} \qquad& \mathrm{tr}({\bf X}) = 1,\\
&...
- 43
2
votes
2
answers
132
views
Invertibility of one matrix constructed by order n subgroup of symmetric group
Let $S_n$ be the symmetric group on $n$ elements $\{ 1,2,\dotsc,n \}$ and $G$ be a subgroup of $S_n$ of order $n$. Denote the elements in $G$ by $\{ \sigma_1,\dotsc,\sigma_n \}$. Let the matrix $A=(\...
- 21
9
votes
1
answer
158
views
Eigenfunctions of the Laplace–Beltrami operator on the coadjoint orbit of $\mathfrak{su}(n)$
$\DeclareMathOperator\SU{SU}$For $\mathfrak{su}(2,\mathbb{C})$, the generic coadjoint orbit is $\mathbb{S}^2$, and the Laplace–Beltrami operator on it is given by
$$
\Delta \equiv \frac{1}{\sin\theta} ...
- 195
2
votes
1
answer
143
views
Inequality for hermitian matrices
Let $p_1$ and $p_2$ be a complete system of orthogonal projections on $\mathbf R^n, n \geq 2$ (i.e., $p_i^2=p_i=p_i^*$ and $p_1+p_2=\bf{1}$) and $S_1, S_2$ be two hermitian operators such that $S_i \...
- 73
4
votes
0
answers
220
views
+50
A question in spin geometry in dimension 8
$\DeclareMathOperator\trace{trace}\DeclareMathOperator\End{End}\DeclareMathOperator\Trace{Trace}$This is to understand a very specific isomorphism in dimension $8$. In dimension $4$ for a spin$^c$ ...
- 954
-3
votes
1
answer
194
views
Bounding a number-theoretic integral
Find a good upper bound on $$\int_1^T\frac{\zeta'(s)}{\zeta(s)\zeta(1-s)}X^sdt,$$ where $s=c+it$ for a constant $c>1$ and $X>0$ is a parameter. If needed, we can assume RH.
My attempt here is ...
- 107
1
vote
1
answer
151
views
Is smoothness preserved under an isometric isomorphism?
Let $(X, \|.\|_1)$ is isometrically isomorphic to $(X, \|.\|_2)$ and $\|.\|_2\leq \|.\|_1$. Assume that $x_0$ is a smooth point of $(X, \|.\|_1)$ and $\|x_0\|_2=1$. According to the definition of a ...
- 113
0
votes
1
answer
121
views
Inequality for commuting hermitian operators
Let $p_1$ and $p_2$ be a complete system of orthogonal projections on $R^n$, $n\geq 2$ (i.e., $p^2_i=p_i=p^*_i$ and $p_1+p_2=\bf{1}$) and $S_1,S_2$ be two commuting hermitian operators on $R^n$ (i.e., ...
- 73
1
vote
1
answer
90
views
Sobolev inequality with weight in the case $1<n\leq p$
Assume that $1<n\leq p$. Does there exist a (non-negative) measure $\mu$ (preferably with some positive density function with respect to the Lebesue measure $dx$) and $q>p$ so that for all $f\in ...
- 459
-5
votes
1
answer
86
views
Why is the second order correction to energy zero for a fully degenerate eigensystem? [closed]
Consider the system given by,
$$ H|n\rangle = E|n\rangle$$
where:
$H$ is the hamiltonian.
$|n\rangle$ is the eigenstate.
$E$ is the energy of the eigenstate.
Using degenerate perturbation theory and ...
- 23
1
vote
0
answers
115
views
How can I derive $V$ from the following equation?
$$
[(U-V)^{-1}X^{T}Y]^{T}\;[(U-V)^{-1}X^{T}Y] = I.
$$
Here $U$ and $V$ are symmetric $d \times d$ matrices; $X=[x_1,x_2,...,x_n]$ is an $n \times d$ data matrix ($n$ is the number of samples and $d$ ...
- 11
-3
votes
0
answers
139
views
A presentation for the group $GL(n,\mathbb{Z}_p)$
Let $n\ge 2$. Let $p$ be a prime and $\mathbb{Z}_p$ denote the finite field with $p$ elements.
I want to know about the presentation for the group $GL(n,\mathbb{Z}_p)$ consisting of its generators and ...
- 103
1
vote
0
answers
91
views
How to optimize parametric information-theoretic bounds?
I am faced with an information-theoretic upper bound, such as
\begin{align}
\sqrt{\alpha'}2^{I_\alpha(X;Y)},
\end{align}
where $I_\alpha(X;Y)$ is the Rényi mutual information with parameter $\alpha>...
- 287
1
vote
1
answer
41
views
Lower spectral radius of matrices with an invariant subspace
Let $\mathcal{A} = \{ A_1, \ldots, A_J, : A_j \in \mathbb{R}^{s\times s}\}$, and define the lower spectral radius (LSR) (aka joint subradius) by
$LSR(\mathcal{A}) = \lim_{n\rightarrow\infty} \inf_{A_{...
- 131
4
votes
0
answers
101
views
Dimension of the intersection of the commuting variety with a particular subspace
Let $\mathcal C$ denote the commuting variety of pairs of matrices in $M_n(\mathbb{C})$, defined as:
$$
\mathcal C = \{ (A, B) \in M_n(\mathbb{C})^2 \mid [A, B] = 0 \}.
$$
It is well known that $\...
- 309
-3
votes
1
answer
46
views
Orthogonal Complement of Orthogonal Complement of a Subset [closed]
The following is an attempt at a proof that $S=(S^\perp)^\perp$ for any $S \subseteq V$. I have reason to believe the conclusion is not true in general but I can not find any errors in this proof. If ...
- 1
0
votes
0
answers
87
views
Is there a name for "applying linear operations to vector sequences from the right"?
Let $v_1,...,v_n\in\Bbb R^d$ be a sequence of vectors. When we say that we "linearly transform" this sequence, we mean that we apply a linear transformation $T\in\Bbb R^{d\times d}$ to each ...
- 13.6k
4
votes
0
answers
103
views
Relationship between characteristic polynomials of a matrix and its adjoint representation
Let $A \in \mathrm{M}_n(F)$ be a matrix over a field $F$. Consider its adjoint representation $\mathrm{ad}_A \in \mathrm{End}(\mathrm{M}_n(F))$, defined by
$$
\mathrm{ad}_A(X) = [A, X] = AX - XA.
$$
I ...
- 309
0
votes
0
answers
118
views
Uncomplete argument in Nishioka book
In Nishioka book "Mahler functions and transcendence" in the proof of Theorem 4.2.1, Nishioka asserts the following: For a matrix $A=(a_{i,j})_{1\le i\le m}$ with coefficients in $K[z]$ ($K$ ...
- 3,996
0
votes
0
answers
46
views
max eigenvalue and schatten-1 norm of depolarizing channel acting on trace-0 Hermitian matrix
Denote $\mathcal{H}^n$ as the $n-$dimension Hermitian matrices. Depolarizing channel $\Delta_p:\mathcal{H}^2\to\mathcal{H}^2$ is defined as $\Delta_p(A)=p\text{ tr }(A)~I/2+(1-p)A$ where $A\in \...
- 91
2
votes
0
answers
59
views
Tensor product of two transcendental flat algebras is not a field?
I'm considering the correctness of the following assertion, which is related to linear disjointness (I'm trying to generalize it to subalgebras), What does "linearly disjoint" mean for ...
- 173
0
votes
0
answers
22
views
Has this notion of "variation along the diagonal of a not-necessarily-smooth function" been studied before?
I am interested in knowing whether something along the lines of the "diagonal variation" defined below has been studied before. In spirit, the basic idea is that it is a kind of ...
- 708