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What are all the complex structures on $\mathbb{R}^2$ which live inside $\mathrm{SL}_2(\mathbb{Z})$?

By "complex structure" I am referring to 2x2 matrices which square to $-\mathrm{Id}_2$. I need to know those with integer entries and determinant equal to 1. Thank you
nayreel's user avatar
-3 votes
0 answers
150 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 ...
SPDR's user avatar
  • 103
4 votes
0 answers
103 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 $\...
darko's user avatar
  • 309
7 votes
1 answer
492 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 \...
Goulifet's user avatar
  • 2,306
8 votes
1 answer
361 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 $i=k$ ...
Richard Stanley's user avatar
0 votes
1 answer
124 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., ...
Svata's user avatar
  • 73
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 ...
Jz Pan's user avatar
  • 173
9 votes
1 answer
296 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 ...
Dima Roytenberg's user avatar
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 \...
Svata's user avatar
  • 73
4 votes
1 answer
254 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$^c$ ...
Partha's user avatar
  • 954
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 ...
Tuh's user avatar
  • 113
-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 ...
user544899's user avatar
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} ...
SCh's user avatar
  • 195
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 ...
P.H.'s user avatar
  • 43
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 \...
qmww987's user avatar
  • 91
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 $K[z]$ ($K$ ...
joaopa's user avatar
  • 3,998
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 $...
abx's user avatar
  • 38k
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>...
Math_Y's user avatar
  • 287
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}\\ ...
Federico Poloni's user avatar
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_{...
tommsch's user avatar
  • 131
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 $d$ ...
dawei's user avatar
  • 11
4 votes
1 answer
175 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,\\ &...
usergh's user avatar
  • 43
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 ...
M. Winter's user avatar
  • 13.6k
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=(\...
lin's user avatar
  • 21
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 ...
darko's user avatar
  • 309
7 votes
2 answers
244 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 &...
Alireza Bakhtiari's user avatar
4 votes
0 answers
291 views

Lower bound on size of the set of sums and differences of non-orthogonal pairs of vectors over finite field

Consider $\mathbb{Z}_m^n$, an $n$-dimensional vector space over $\mathbb{Z}_m$. For two sets of vectors $P = \left\{ p^i \right\}$ and $Q = \left\{ q^j \right\}$ and a skew-symmetric matrix $S_{ij}=\...
EvgeniyZh's user avatar
0 votes
1 answer
48 views

How to efficiently replace the Paterson-Stockmeyer algorithm and reduce non-scalar multiplications

Paterson-Stockmeyer algorithm If we need to compute a high-degree polynomial expression, such as: $$ P(y) = \sum_{k=0}^{B} a_k y^k $$ the Paterson-Stockmeyer algorithm can process the powers in ...
AC.PR's user avatar
  • 3
0 votes
1 answer
157 views

Can the derivative of eigenvectors with respect to its components be taken as zero if all eigenvalues are equal?

I want to ask a couple of follow up questions to the question answered on the thread "Derivative of eigenvectors of a matrix with respect to its components". I noticed that in the accepted ...
user544899's user avatar
3 votes
1 answer
111 views

Generalization of a result of Kostant related to Gauss decomposition and Toda lattices

I found myself needing a generalization of a result of Kostant in his famous paper B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math, Volume 34, 1979, ...
Three aggies's user avatar
1 vote
3 answers
257 views

Eigenvalues of positive matrices in $\mathrm{SL}(d,\mathbb{Z})$

Let $A\in\operatorname{SL}(d,\mathbb{Z})$ be an irreducible positive matrix, i. e. $A=(a_{i,j})_{1\leq i,j\leq d}$ with $a_{i,j}\in\mathbb{Z}_{>0}$. From the Perron-Frobenius theorem, we know that $...
Yi SHI's user avatar
  • 11
1 vote
0 answers
67 views

System of linear diophantine equations with many small solutions?

Let $n$ be positive integer, $k$,$B$ fixed positive integers. Let $f_i(x_1,x_2...x_n)$ be a system of $n-k$ linearly independent linear equations over the integers. Let $S(f_i,k,B)$ be the set of ...
joro's user avatar
  • 25.4k
1 vote
1 answer
76 views

Determinant formula for a certain parametrized M-matrix

Let $P_{ij}$ be variables, and let $A \in \mathbb{R}^{n\times n}$ be the matrix defined by $$ A_{ij} = \begin{cases} -P_{ij} & i \neq j,\\ P_{i1} + P_{i2} + \dots + P_{in} & i=j. \end{cases} $$...
Federico Poloni's user avatar
2 votes
1 answer
107 views

Linear automorphism preserving a cone

Let $V$ be a finite-dimensional real vector space, and let $C\subset V$ be a closed convex cone, not contained in a hyperplane, and such that $C\cap(-C)=\{0\} $. Let $n$ be a nilpotent endomorphism of ...
abx's user avatar
  • 38k
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 $$ f_n(X) = \mathbb{E} \langle (X^{-1} + gg^T)^{-1} u, u\rangle, $$ ...
Drew Brady's user avatar
5 votes
0 answers
231 views

Avoiding Cartan subalgebra in a Lie algebra

Let $G$ be a simple complex algebraic group acting on its Lie algebra $\mathfrak{g}$ via the adjoint representation. What is the largest integer $d$ such that every subspace $U \subseteq \mathfrak{g}$ ...
darko's user avatar
  • 309
2 votes
0 answers
104 views

Find an $a \times m$ submatrix of an $n \times m$ matrix with smallest rank

Given a matrix $n \times m$, I want to find the submatrices $a \times m$ by selecting $a$ columns such that their rank is minimal. Can this problem be solved efficiently?
Alm's user avatar
  • 1,207
0 votes
1 answer
98 views

Only special permutations result in a constant expression when permuting coefficients in a sum involving binomials?

Fix $n\geq 1$ and let $p_k(x) := x^k(x-1)^{n-k}$. Suppose $\pi$ is a permutation on $\{0,1,\dotsc,n\}$, such that $$ \sum_{k=0}^n (-1)^k \binom{n}{k} p_{\pi(k)}(x) \text{ is a constant}. $$ Must it be ...
Per Alexandersson's user avatar
11 votes
2 answers
387 views

Bounds for the difference in the number of ones in $M$ and $M^{-1}$

If $M$ is a full rank $n$ by $n$ binary matrix over $\mathbb{F}_2$, how much larger or smaller can the number of $1$s in $M^{-1}$ be, compared to the number of $1$s in $M$? Clearly the identity matrix ...
Simd's user avatar
  • 3,377
9 votes
3 answers
1k views

Examples of combinatorial problems where the only known solutions, or most "natural" solutions, use representation theory?

In Solution of two difficult combinatorial problems with linear algebra, Robert Proctor presents two simply stated combinatorial problems, and gives solutions to them using a linear algebraic approach ...
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 ...
Isaac's user avatar
  • 3,477
0 votes
2 answers
97 views

Optimization algorithms for Kronecker approximation of high-dimensional covariance matrices

I'm working with a high-dimensional covariance matrix and exploring Kronecker product approximations to make it computationally manageable. Here's the setup: I have a graph $G$ represented by a $D\...
JJbox's user avatar
  • 1
5 votes
1 answer
539 views

Under what circumstances Is a symmetric matrix representable as a Coulomb matrix?

Question: I am exploring a neural network architecture inspired by physical interactions, where each neuron has associated "mass" and "position" vectors. The weight matrix between ...
mathoverflowUser's user avatar
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 ...
dzp's user avatar
  • 21
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 $f$ ...
SmileyCraft's user avatar
7 votes
1 answer
292 views

Existence of matrix diagonalizing $x A + y B$ for all $x, y$ and independent of $x, y$

Let $A_1, A_2$ and $B_1, B_2$ be real symmetric matrices. Suppose $x A_1 + y B_1$ is cospectral with $x A_2 + y B_2$ for all real numbers $x, y$. Is it true that there exists a fixed orthgonoal matrix ...
MMM's user avatar
  • 325
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}{...
Kawa's user avatar
  • 11
4 votes
1 answer
236 views

If a lattice can be embedded into $\mathbb Q^n,\langle-1\rangle^n$, then can it be embedded into $\mathbb Z^n,\langle -1 \rangle^n$?

Given a graph with negative integers on each vertex $\Gamma$ there is a corresponding intersection lattice denoted $Q_\Gamma$, a free $\mathbb Z$ module generated by the vertices, endowed with a ...
Márton Beke's user avatar
0 votes
0 answers
50 views

Eigenvalues of functions on finite discrete sets

Suppose I have an arbitrary function on a finite and discrete set $S$ defined as $$f: S \times S \to \mathbb{C}^{|S|\times |S|}.$$ The $|S| \times |S|$ matrix $M$ is defined as $$(M)_{ij}=f(s_i, s_j) \...
aa190903's user avatar
4 votes
2 answers
587 views

Is this function injective?

For all given ordered lists $$\mathcal A=\big\{\{a_\mu\mid\mu=1,\cdots,N\}\mid\forall\mu,\nu> \mu,\ a_\mu > a_\nu\big\},$$ the function on the quotient space $$ G_\mu(a+\mathbb R: \mathcal A / \...
Gnaphalium's user avatar

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