All Questions
6,292 questions
0
votes
1
answer
776
views
Highest weights of irreducible components of tensor product of irreducible sl(3)-module [closed]
I am study the representation theory of $sl(3)$ and I have a question about the tensor representation of irreducible $sl(3)$-modules as follows:
For each weight $\mu$, let $L(\mu)$ be the irreducible ...
- 11
4
votes
1
answer
331
views
Non-orthogonal vectors cosine-enhancing transformation
Let consider a real vector space $\mathbb{R}^n$ of dimension $n$, where $\langle \cdot, \cdot\rangle$ and $||\cdot ||$ are the standard inner product and $\ell_2$ indeced norm.
Does there exist a ...
- 424
-1
votes
1
answer
681
views
Is there such a thing [closed]
For any $U_{i}\in\mathcal{U}\left(4\right)$, $1\le i\le5$, are there
$W\in\mathcal{U}\left(4\right)$ and nontrivial $\left(x_{1},x_{2}\right)\in\mathbb{R}^{2}$, such that $\mbox{tr}\left(U_{i}\mbox{...
- 53
7
votes
2
answers
2k
views
Does anyone know what is the right reference for the following simple lemma from harmonic analysis?
The lemma says that given $\lambda\geq 1$, $p\geq 1$, $a_j\geq 0$, for a collection of balls $\{B_j\}_{j\in\mathbb{N}}$ in $\mathbb{R}^n$, it holds
$$\bigg\|\sum_j a_j\chi_{\lambda B_j}\bigg\|_p\leq C(...
- 1,881
13
votes
2
answers
2k
views
An alternative proof of the Łojasiewicz inequality
Is there a "brute force proof" of the Łojasiewicz inequality? By "brute force" I mean a proof without introducing the machinery of semianalytic sets and so on but only using elementary results (i.e., ...
- 1,727
11
votes
2
answers
2k
views
History of Jordan Canonical Form?
Can anyone suggest a reference that discusses the history of the Jordan canonical form? In particular, I am interested in:
When and how was it first stated? (I understand it was independently stated ...
- 3,761
2
votes
1
answer
387
views
Is this projective variety empty?
Let $\mathbf{a}\in\mathbb{R}^{4}$ and the entries of a $3$ by $4$ real matrix $A$
be the variables of the polynomial equations, $\det\left(A\right)=0$, $S_{1},S_{2}\subseteq\left\{ 1,2,3,4\right\}$
...
- 31
0
votes
1
answer
268
views
Nonnegative Matrix
Let $A=E+\sqrt{-1}B$, where $E=diag\{0,1,\cdots,1\}$, $B$ is a real symmetric matrix. Let $A^*$ denote the adjoint matrix of $A$, i.e. $AA^*=\det A\cdot I$. I hope the real part of adjoint matrix ${\...
- 303
3
votes
3
answers
241
views
For centralizer subgroups, is the endomorphism ring of a restriction generated by endomorphisms and the centralized element?
In some recent doodlings, I got myself to the point where what I was trying to understand would work out if the following claim were true:
Let $G$ be a group, $g\in G$, and $\rho:G \to \...
- 54.6k
9
votes
1
answer
598
views
Trouble with Jordan form of the truncated Carleman-matrix for $\sin(x)$ as size $n$ goes to infinity
I'm currently trying to get familiar with the Jordan normal form for matrices; and after some example I ask for the possible Jordan-form for the Carleman matrix for the function $f(x) = \sin(x)$ when ...
- 5,342
0
votes
1
answer
151
views
Measuring the extent of entanglement in tensor products
Is there a non-negative integer valued grading function on the tensor product of two Hilbert spaces which measures the extent of entanglement ?
2
votes
1
answer
1k
views
Form of a block upper triangular matrix of finite order
If I take a diagonalizable block upper triangular matrix whose diagonal blocks are of finite order, is it true that away from the leading block diagonal, the matrix is zero?
I think the statement is ...
- 21
4
votes
1
answer
116
views
Set of orthogonal simplexes or partial mutually unbiased bases
I am interested in the existence of a set of vectors $\{ v_{ij} \}_{ij} \subseteq \mathbb{C}^N$ for $i \in \{1,\dots,N\}$, $j \in \{1,\dots,N+1\}$ such that $\left\vert v^*_{ij} v_{ij'} \right\vert = ...
- 298
3
votes
1
answer
427
views
Ranks of higher incidence matrices of designs
In 1978 Doyen, Hubaut and Vandensavel proved that if $S$ is a Steiner triple system $S(2,3,v)$ then the $GF(2)$ rank of its incidence matrix $N$ is
$$
Rk_{2}(N)=v-(d_{p}+1),
$$
where $d_{p}$ is the ...
- 6,962
0
votes
2
answers
218
views
Does there exist a linear equation system with a NAND-like behavior? [closed]
Does there exist an equation system with only linear equalities and inequalities and exhibits a NAND-like behavior?
By 'NAND-like' I mean, among all the variables in the system there are 3 vars $x_1$, ...
- 117
1
vote
0
answers
111
views
Solve a linear equation with many variables using only 1 and -1
For a program I am writing, I would find it useful to find the least possible positive solution to a linear equation, using only -1 and 1 for roots.
For example...
...
- 111
4
votes
3
answers
5k
views
Online estimation of covariance matrix
I am trying to dynamically estimate the (low-dimensional) covariance matrix ${\mathbb E}[{\bf x}_t{\bf x}_t^\top]$ of a stream of data points ${\bf x}_t\in{\mathbb R}^N$ online, without any memory. ...
- 51
2
votes
2
answers
825
views
Designing almost orthogonal vectors in a deterministic manner
Consider the vector space $\mathbb{R}^n$, the standard inner product $\langle \cdot,\cdot \rangle:\mathbb{R}^n\times\mathbb{R}^n\rightarrow \mathbb{R}$, and some $0<\epsilon\leq \frac{1}{\sqrt{n}}$....
- 163
1
vote
1
answer
83
views
Augmenting orthonormal system into complete orthonormal system in a numerically stable way
Let us suppose we have a, say, 10 dimensional real space with 3 orthogonal unit vectors given. How do I complete this orthonormal system with 7 additional vectors into a complete ONS in a way that is ...
- 45
4
votes
1
answer
513
views
Checking irreducibility
This is related to this question. Suppose I have an $n$-dimensional representation of a finitely generated group, and I want to know whether it is absolutely irreducible. This can, of course, be done ...
- 96.4k
3
votes
1
answer
427
views
Graph of Grassmannian
Let p be an integer, and let G be the graph $(V=Gr(k,\mathbb{F}_q ^n),E)$
where: $Gr(k,\mathbb{F}_q ^n)$ is the set of all subspace of $\mathbb{F}_q$ of dimension k, and $E=\{ W_1,W_2 \in V | W_1\...
- 33
1
vote
1
answer
227
views
Do the cycles containing a fixed edge generate the cycle space of a graph?
Let $G$ be a $2$-connected graph and for $e \in E(G)$ denote by $\mathcal{C_e}$ the set of all cycles of $G$ containing the edge $e$.
For what set of edges does $\mathcal{C_e}$ contain a basis of the ...
- 1,034
11
votes
4
answers
5k
views
The derivative of the Cholesky factor
Let $A$ be a symmetric, positive definite $p\times p$ matrix, and let $f(A)$ be its Cholesky factor. That is, $f(A)$ is a lower triangular $p\times p$ matrix such that $A = f(A) f(A)^{\top}$. I am ...
- 620
2
votes
0
answers
529
views
Inverting a matrix with entries equal to positive or negative infinity
I would like to define an inverse on matrices whose entries may be positive or negative infinity.
To formulate my problem precisely, suppose that I have a matrix $A$ and another matrix $B$. How do I ...
- 45
5
votes
1
answer
993
views
Full-rank rectangular matrices over GF(2)
Given positive integers $k$, $m$, $n$, let $A$ be an $m \times n$ matrix over $GF(2)$ constructed as follows. Let $X_1, \ldots, X_m$ be independent random subsets of $\{1,\ldots,n\}$ with cardinality ...
- 54.2k
3
votes
3
answers
1k
views
Are the finite dimensional von Neumann algebras, singly generated?
Let $\mathcal{M}$ be a finite dimensional von Neumann algebra, then :
$$\mathcal{M} \simeq \bigoplus_i M_{n_i}(\mathbb{C})$$
Question : Is it singly generated (as von Neumann algebra)? how ?
...
9
votes
1
answer
420
views
Coherence between different ranking methods of a graph's vertices
Given a (connected) graph $G$ it is natural to want to rank its vertices, with the more "central" vertices ranked higher.
Two natural ways of doing it are:
By the degrees.
By the entries in a Perron ...
- 6,962
3
votes
1
answer
494
views
A question on Grassmannian
Let $V$ be the space of $4$ by $4$ Hermitian matrices, that
is a vector space of dimension $16$ over $\mathbb{R}$. Is the uniform
measure of
$$
\left\{ W\in Gr\left(5,V\right):W \text{contains no ...
- 119
3
votes
1
answer
966
views
A submanifold of the space positive definite matrices
Consider the space of $n \times n$ positive definite symmetric matrices and let $\Sigma$ be one such matrix. We make this space into a Riemannian manifold $M$ by means of the metric $$ds^2=tr(\Sigma^{-...
- 283
8
votes
1
answer
541
views
A question on eigenvalues
Let $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$, $A_{5}$ be linearly independent Hermitian matrices in the the space of $6$ by $6$ Hermitian matrices as a vector space over $\mathbb{R}$. Does there always ...
- 119
2
votes
1
answer
99
views
Rings in which every J-matrix is non-singular
Let $R$ be a ring with identity. A matrix $A=[a_{ij}] \in M_n(R)$ is called a J-matrix if for any $i$, $a_{ii} \not \in J(R)$ but for any $i \not = j$, $a_{ij} \in J(R)$. Now suppose that every J-...
- 123
6
votes
0
answers
318
views
Variant of orthogonal Procrustes problem
The orthogonal Procrustes problem seeks a matrix $M$ that minimizes $||AM-B||_F$ subject to $M^TM=I$, where $M$ is $d\times d$ and both $A$ and $B$ are $n\times d$. Geometrically, $M$ rotates a set of ...
- 61
10
votes
2
answers
2k
views
Characteristic polynomial of exterior power
Suppose $f$ is a linear map, and consider $\Lambda^k f$ as the usual exterior power of $f$ (if you prefer matrices, it is a matrix whose entries are the $k\times k$ minors of $f.$) The coefficients of ...
- 96.4k
5
votes
1
answer
420
views
Which positive definite symmetric matrices have solvable characteristic polynomial?
I am interested in the structure of the space of $n \times n$ positive definite symmetric matrices with rational entries whose characteristic polynomials are solvable (i.e. the Galois group is ...
- 29.2k
27
votes
2
answers
1k
views
Continuous functions $f$ with $f(A)$ linearly independent when $A$ is independent
Is there any characterization of continuous functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ such that for any linearly independent set $A$ (over the rationals) $f(A)$ is also linearly independent ?
- 447
3
votes
0
answers
264
views
Does the product of principal sines between subspaces satisfy the triangle inequalilty?
As we know that the volume of a matrix $X$ is defined as $\sqrt{\det(X^TX)}$, if we consider the volume of two matrices $X$ and $Y$, with $X,Y \in \mathbb{R}^{N\times d},d<N,\dim(X)=\dim(Y)=d$, ...
- 131
9
votes
3
answers
4k
views
Spectra of spatial and temporal covariance matrices
Suppose $(x_i(t))$ is a $n$-dimensional time-series, where $t$ is an integer between $1$ and $T$ (time is discrete) and $i$ an integer between $1$ and $n$, and I assume $n<T$. From this time-series,...
- 840
-2
votes
1
answer
871
views
Rank of a random matrix
Let $x$ a random Gaussian vector of size $n$ with i.i.d coefficients $N(0,1)$. Let $J$ a random matrix with i.i.d coefficients $N(0,\sigma^2/n)$ where $\sigma \in [0,1]$. For any integer T>n, define:
$...
- 840
4
votes
2
answers
401
views
Variety determined by interior product of the determinant?
Let $\Lambda^k(V)$ be the space of alternating $k$-linear tensors on $V$. Consider the map $f: \left(\mathbb{R}^n\right)^{n-k} \to \Lambda^k(\mathbb{R}^n)$ given by $\left(v_1,v_2, ..., v_{n-k}\right)...
- 12.1k
2
votes
2
answers
765
views
Why eigenvectors optimize this orthogonally constrained nonlinear minimization problem?
Given a $p \times p$ positive definite matrix $\Sigma$, why eigenvectors of $\Sigma$, stacked as columns of a matrix $R \equiv [r_1 \, r_2 \, \ldots \, r_p]$, optimize the following orthogonally ...
- 362
9
votes
1
answer
3k
views
Frobenius-Perron eigenvalue and eigenvector of sum of two matrices
Suppose that I have two positive matrices, $A$, and $B$, and I know their Frobenius-Perron eigenvalues ($\lambda_A$, $\lambda_B$) and eigenvectors ($v_A$, $v_B$). I'm interested in what I can say ...
- 403
4
votes
0
answers
390
views
Fully Homomorphic Error Correction?
Consider a field $F$. Suppose we have two vectors $a,b\in F^n$, and an invertible matrix $G\in F^{n\times n}$. Let $c\in F^n$ be the point-wise product of $a$ and $b$, that is, $c_i=a_ib_i$. Let $x=...
- 3,979
0
votes
3
answers
2k
views
Multiplicative functions $\phi : M_n(F) \longrightarrow F$ with $\phi(I) = 1$
Let $F$ be an infinite field and let $f \in F[x_{11},x_{12},...,x_{nn}]$ be an arbitrary polynomial in $n^2$ variables. Consider the function $\phi : M_n(F)\longrightarrow F$ defined by $\phi((a_{...
- 447
0
votes
0
answers
117
views
"Almost orthogonalizing" matrices using a signature matrix
Suppose $A$ and $B$ are two real symmetric $n \times n$ matrices (If simpler, consider $A$ and $B$ to be 0/1 matrices, say, adjacency matrices of d-regular graphs).
Then $||AB||_{op} \leq ||A||_{op}||...
- 949
12
votes
1
answer
4k
views
How to check whether a positive integer can be written as linear combination of given others, where all coefficients are positive?
Let $n$, $k$ and $m_1, \dots, m_k$ be positive integers. Which is the most efficient
algorithm to find out whether there are positive integers $a_1, \dots, a_k$ such that
$n = \sum_{i=1}^k a_i m_i$?
...
- 241
0
votes
1
answer
164
views
On separable field extensions [closed]
Let $F\subseteq K$ be a finite separable field extension with $a_1,..., a_n$ an $F$-basis for $K$. Is it true that the matrix $A := [\mbox{tr}(a_ia_j)]$ is non-singular ?
- 447
1
vote
1
answer
246
views
Eigenvalue problem with quadratic constraints
$\circ$ Consider the following eigenvalue problem : $$Ax=\lambda x \hspace{0.5cm} (1)$$
where matrice $A \in \mathbb{R}_{n \times n}$ is a positive semi-definite with eigenvectors $x = (x_{1},x_{2},.....
- 45
5
votes
2
answers
481
views
Decomposition of a semi-definite matrix into sums of sparse semi-definite matrices
I'll first provide the background.
Let $x\in\mathbb{R}^N$ be decomposed into $n$ non-overlapping blocks of variables
$x^{(1)},\ldots,x^{(n)}$.
We say that $f:\mathbb{R}^N\rightarrow\mathbb{R}$ is ...
- 51
6
votes
1
answer
305
views
For a set of matrices $S$, find $X$ such that the elements of $SX$ commute
Let $S := \{A_0, A_1, \dots, A_d\}$, where $A_k \in \mathbb{C}^{n \times n}$, be a set of (generally noncommuting) matrices. I am interested in finding a nonsingular $X \in \mathbb{C}^{n \times n}$ ...
- 300
2
votes
0
answers
58
views
Moments of matrix multiplied with diagonal matrix with Gaussians
I have a circulant Toeplitz matrix $T$ and a diagonal matrix $D$ with IID zero mean complex Gaussians with circular symmetry along its main diagonal.
How do I compute the moments $E[(D^HTD)^n]$?
The ...
- 21