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On tensor products of "generic" vectors

Suppose that $x_1,\ldots,x_n$ are $n$ vectors in $\mathbb{R}^m$ (where $m<n\leq m^2$) such that any subset of $m$ of them are linearly independent (i.e., they are "generic"). Now, form the $m^2\...
Skoro's user avatar
  • 168
4 votes
1 answer
2k views

Determinant and symmetric power

Let $V$ be a vector space over some field $k$ and $T \in \mathrm{GL}(V)$. Then, we can view $T\in \mathrm{GL}(\mathrm{Sym}^k(V))$ where $\mathrm{Sym}^k(V)$ denotes the $k^\mathrm{th}$ symmetric power ...
Brian's user avatar
  • 1,510
4 votes
1 answer
494 views

Characterization of all-orthogonal tensors

In the paper [1], it is proven in Theorem 2 that any $n$-tensor $\mathcal{A}\in\mathbb{R}^{d_1\times...\times d_n}$ can be decomposed as $$ \mathcal{A}=\mathcal{S} \times_1 U_1 ...\times_n U_n $$ ...
Bonnevie's user avatar
4 votes
1 answer
1k views

Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix $\mathbf{A}$. First, let's define two matrices: ...
anderstood's user avatar
4 votes
2 answers
433 views

What is the current status of representation theory of $n$-ary groups in terms of hypermatrices?

An $n$-ary group is a generalization of the usual concept of a group where the binary operation (2-argument operation) is instead an $n$-ary ($n$-argument) operation. More info here on Wikipedia. I ...
Sidharth Ghoshal's user avatar
4 votes
0 answers
124 views

Distributive lattices with periodic Coxeter matrix

Let $L$ be a finite distributive lattice and $U$ its incidence matrix with entries $u_{i,j}=1$ iff $i \leq j$ and $u_{i,j}=0$ else. Then $U^{-1}$ is the Moebius matrix of $L$ and $C_L:=- U^{-1} U^{T}$ ...
Mare's user avatar
  • 26.5k
4 votes
3 answers
381 views

Is a simple graph the "sum" of a partial order and its dual?

A "$n$-order matrix" $T\in M_n(\mathbb F_2)$ is a matrix such that there exists a partial ordered relation $\leq_T\subset [1,n]^2$ such that : $T_{ij}=1\Leftrightarrow i\leq_T j$ (where $T_{ij}$ is ...
jcdornano's user avatar
  • 469
4 votes
2 answers
818 views

Double orthogonal complement of a finite module

Crossposted from math.stackexchange since I'm not getting any answer. Let $W$ be the finite $\mathbb{Z}$-module obtained from $\mathbb{Z}_q^n$ with addition componentwise where $\mathbb{Z}_q$ is the ...
Carl's user avatar
  • 141
4 votes
1 answer
410 views

power log distance between matrices

In this thesis, Pedro Freitas discusses the properties of distance functions on matrices defined by $d_p(A, B) = (\sum (\log (\sigma_i(A^{-1} B)))^p)^{1/p}.$ Here $\sigma_i$ are the singular values of ...
Igor Rivin's user avatar
  • 96.4k
4 votes
0 answers
192 views

Forcing scalar products to avoid prescribed values

Let $p$ be a prime, and $n\ge 1$ an integer number. Suppose that the (not necessarily distinct) vectors $v_1,\dotsc,v_N \in{\mathbb F}_p^n$ satisfy the following condition: \begin{gather} \text{For ...
Seva's user avatar
  • 23k
4 votes
4 answers
3k views

The multiplicity of the max eigenvalue in matrix multiplication

Suppose that eigenvalues of two real square matrix $A$ and $B$ are $1 = \lambda^A_1 > \lambda^A_2 \geq \ldots \geq \lambda^A_n > 0 $ and $1 = \lambda^B_1 > \lambda^B_2 \geq \ldots \geq \...
David's user avatar
  • 41
4 votes
2 answers
311 views

Solving $\text{trace}\left[\left(I + pY\right)^{-1} \left(I - p^{2}Y\right)\right] = 0$ for scalar $p$

For given $n\times n$ real symmetric positive definite matrices $X_{1}, X_{2}$, let $Y := X_{1}^{-1}X_{2}$, and let $I$ be the identity matrix. I would like to solve the following equation for the ...
Abhishek Halder's user avatar
4 votes
1 answer
1k views

Determinant involving traceless unitary hermitian matrices

Let $S$ be the set of complex $N\times N$ matrices that are traceless, unitary and hermitian. A friend asked me the following question, motivated by a problem in condensed matter physics: Is it ...
thedude's user avatar
  • 1,549
4 votes
1 answer
153 views

Linear maps preserved by algebra automorphisms

Let $A$ be a finite dimensional , associative, unital $F$-algebra, where $F$ is a field. Let $s_A:A\to F$ be an $F$-linear map. Now consider an arbitrary field extension $K/F$, and define $s_{A\...
GreginGre's user avatar
  • 1,766
4 votes
1 answer
294 views

What is this matrix decomposition called and does it exist always?

Given a rank $2r$ matrix $M\in\Bbb Q^{n\times n}$ can we find two matrices $M_+\in\Bbb Q_{\geq0}^{n\times n}$ and $M_-\in\Bbb Q_{\geq0}^{n\times n}$ each of rank $r$ such that $M=M_+-M_-$ holds? ...
Turbo's user avatar
  • 13.9k
4 votes
1 answer
351 views

Maps between products of symmetric powers

This question might be too elementary but it arises naturally as a part of a more complicated computation and I struggle to find the answer. Let $V$ be an $n$-dimensional complex vector space. ...
clementine's user avatar
4 votes
1 answer
875 views

comparing norms of tensor product of two Hilbert spaces

Suppose $H_1$ and $H_2$ are two Hilbert spaces with dimension $n$ and $m$, for $ x \in H_1 \otimes H_2$ consider $$\|x\|_\pi = \inf \left\{ \sum_{i=1}^n \|a_i\| \|b_i\| : x = \sum_{i} a_i \otimes b_i ...
user82336's user avatar
4 votes
1 answer
174 views

A map into a Hilbert space with prescribed orthogonality

Let $X$ be a locally compact separable metric space, and let $L:X\times X\to \mathbb{C}$ be continuous and such that $L(x,x)=1$ and $L(y,x)=\overline{L(x,y)}$, for every $x,y$. Does there always ...
erz's user avatar
  • 5,529
4 votes
2 answers
239 views

Distribution of $0$-$1$ matrices

Consider $n\times n$ matrices with entries in $\{0,1\}$. The determinants of these ranges from $0$ to the Hadamard bound $\frac{(n+1)^{\frac{n+1}2}}{2^n}$. Assume $n$ is large enough. What does the ...
Lewi_Sol's user avatar
  • 309
4 votes
1 answer
535 views

A Question on Koszul duality and $B(\infty)$ structures on $HH^*$

The following theorem is known from a paper "Duality in Gerstenhaber Algebras" by Felix, Menichi, Thomas. Given a simply connected space X of finite type. There is an equivalence of Gerstenhaber ...
Daniel Pomerleano's user avatar
3 votes
0 answers
384 views

Systems of linear octonionic equations

Is there theory of determinants, rank of matrices and systems of linear equations with octonionic coefficients? Does anybody could indicate references? I want to know mainly does there exist a ...
rts's user avatar
  • 75
3 votes
0 answers
207 views

On a variation of the Vandermonde matrix

The ubiquitous Vandermonde matrix, of entries $(x_i^{j-1})_{i,j}^{1,n}$, and its determinant $$\prod_{i<j}^{1,n}(x_j-x_i)$$ have found many utilities in Combinatorics and Physics, among other ...
T. Amdeberhan's user avatar
3 votes
0 answers
237 views

Multi-dimensional permanent of structured tensor

I am facing the multidimensional permanent \begin{equation} \text{perm}(W) = \sum_{\sigma, \rho \in S_n} \prod_{j=1}^n W_{j, \sigma_j, \rho_j } \end{equation} of a 3-tensor $W_{j,k,l}$ of ...
Malte's user avatar
  • 93
3 votes
1 answer
165 views

The inverse of a symbolic matrix (with reciprocal binomials) has Laurent entries

Recalling the $q$-binomials (Gaussian polynomials). Let $[n]_q!=\prod_{j=1}^n\frac{1-q^j}{1-q}$ and $\binom{n}k_q=\frac{[n]_q!}{[k]_q!\cdot[n-k]_q!}$. Now, consider the $n\times n$ matrix $\mathbf{M}...
T. Amdeberhan's user avatar
3 votes
4 answers
4k views

Existence of nonnegative solutions to an underdetermined system of linear equations

Similar questions have been asked elsewhere, but I think this is sufficiently different to warrant a new post. I have a particular matrix $A$ and would like to know when the system $Ax = 0$ has at ...
bandini's user avatar
  • 491
3 votes
1 answer
415 views

Inverse of block matrix

Let $V$ be a finite-dimensional vector space and consider the space $X=V\times V\times V\times V.$ Consider the block matrix $$A = \begin{pmatrix} A_1 & A_2 \\ A_2^* & -A_1\end{pmatrix}$$ ...
Kung Yao's user avatar
  • 192
3 votes
0 answers
202 views

Difficult Gaussian-sum inequality for large random Bernoulli-Toeplitz matrices

I have come across the following problem in an attempt to prove an entropy bound for large random Bernoulli-Toeplitz matrices (Conjecture 1 on p. 16 of this preprint by Clifford et al. 2015), which is ...
Dierk Bormann's user avatar
3 votes
1 answer
399 views

Maximal common isotropic subspace for a finite family of skewforms

Let $V$ be a vector space of dimension $n$ over a field $F$. An alternating bilinear form $\alpha\colon V \times V \rightarrow F $ will be called a skewform. A subspace $W$ is isotropic for $\alpha$...
Sky's user avatar
  • 923
3 votes
0 answers
130 views

Where does this identity involving sums of Hankel-like determinants over partitions come from?

For a partition $\lambda=( \lambda_1,\dots,\lambda_n)\vdash n$ with $\lambda_1\ge\dots\ge\lambda_n\ge0$ and any function $f:\mathbb Z\to\mathbb C$, define a Hankel-like $n\times n$ matrix $$M_f(\...
Wolfgang's user avatar
  • 13.4k
3 votes
1 answer
325 views

Reference on completely positive maps which are isometries

Let $\Phi:\mathcal{L}(H)\rightarrow \mathcal{L}(K)$ be a completely positive map sending positive self-adoint operators on a finite-dimensional Hilbert space $H$ to positive self-adoint operators on a ...
Stefano Gogioso's user avatar
3 votes
0 answers
225 views

Eigenvalues of Hadamard product of two Wishart-type matrices

Given two independent Gaussian matrices with i.i.d. entries: $A\in\mathbb{R}^{n\times p}$ and $B\in\mathbb{R}^{n\times q}$, where and $A_{i,j},B_{i,j}\sim\mathcal{N}(0,1)$. Assume that $\max(p,q)<n....
M-Brust's user avatar
  • 31
3 votes
1 answer
784 views

Expected number of random binary vectors so that the form a basis

I would like to compute the expected number of vectors in $\mathbb{F}_2^n$ we need to draw (following a uniform distribution) so that they form a basis of $\mathbb{F}_2^n$, i.e., that we have $n$ ...
ocalex86's user avatar
3 votes
1 answer
712 views

Relation between eigenvalues and the gram matrix

We have matrix data $X$ which is $n\times d$. We use the covariance matrix/ design matrix/ gram matrix $X^T X$ to perform least-squares/ PCA. I compute the eigen basis representation of said matrix $$...
rostader's user avatar
  • 215
3 votes
0 answers
2k views

Eigenvalue Problems with Linear Constraints

The motivation for this problem comes from trying to develop a simple way to decompose domains into non-overlapping subdomains to solve for the eigenvalues of some differential operator. The idea is ...
Ryan Thorngren's user avatar
3 votes
2 answers
5k views

Decomposition of Matrices in Semisimple and Nilpotent Parts

I asked this question in https://math.stackexchange.com/questions/204115/decomposition-of-matrices-in-semisimple-and-nilpotent-parts ​​but remains unanswered. For any matrix $A\in M_n(\mathbb F)$, ...
Miguel's user avatar
  • 545
3 votes
1 answer
164 views

A system of linear equations related to the geometry of a finite plane

Let $\mathcal L$ denote the set of all lines in $\mathbb F_p^2$ parallel to one of the lines $$ X:=\{(x,0)\colon x\in\mathbb F_p \}, \ Y:=\{(0,y)\colon y\in\mathbb F_p \}, \ Z:=\{(z,z)\...
Seva's user avatar
  • 23k
3 votes
0 answers
234 views

Hurwitz–Radon problem for $ \mathbb{Q} ^{n} $

What is the maximal number of orthogonal operators $ A _{1} , \dotsc, A _ {m} $ in $ \mathbb{Q}^{ n } $ satisfying the relations $ A_{i}^{2} = - I $ and $ A_{i}A_{j} + A_{j}A_{i} = 0 $ for $ i \neq ...
Sky's user avatar
  • 923
3 votes
4 answers
367 views

Prove that $(v^Tx)^2−(u^Tx)^2\leq \sqrt{1−(u^Tv)^2}$ for any unit vectors $u, v, x$

I believe I found a complicated proof by bounding the spectral norm $||uu^T-vv^T||^2_2:=\max_{||x||=1}|(u^Tx)^2-(v^Tx)^2|$. Using the fact that $dist(x,y):=\sin|x-y|$ is a distance function over unit ...
Dan Feldman's user avatar
3 votes
1 answer
272 views

Enumerating possible number of satisfied linear equations

Consider a system of linear equations of variable $x=(x_1,\cdots,x_n)$ where each $x_i\in\{ 0,1,\cdots,L-1 \}$. Clearly, there are $\frac{n(n-1)}{2}$ number of equations in the system. $$x_i-x_j=0, \ \...
tony's user avatar
  • 405
3 votes
1 answer
1k views

Symplectic block-diagonalization of a complex symmetric matrix

This is a follow-up question to the one asked here: Given a complex symmetric $2n\times2n$-matrix $A$, i.e., $A\in \mathbb{C}^{2n\times2n}$ with $A = A^T$. Is it possible, to block-diagonalize $A$ ...
Fabian's user avatar
  • 290
3 votes
2 answers
203 views

Recovering a set from its projections in varying coordinate systems - a projection hull?

Let me describe the simplest non-trivial case of what I have in mind. Let $V$ be a 2-dimensional $\mathbb{R}$-vector space and fix an isomorphism $V \cong \mathbb{R}^2$, where $\mathbb{R}^2$ is ...
M.G.'s user avatar
  • 7,127
3 votes
0 answers
263 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 $X,Y \in \mathbb{R}^{N\times d},d<N,\dim(X)=\dim(Y)=d$, ...
Helmholz's user avatar
  • 131
3 votes
1 answer
1k views

Number of full-rank binary matrices with no rows of weight 1

For $m > n$, I want to calculate the number of binary matrices with $m$ rows and $n$ columns for which two conditions hold: the rank of a matrix is $n$ (i.e. it is full-rank, as $m > n$); there ...
Yauhen Yakimenka's user avatar
3 votes
2 answers
673 views

Asymptotic number of invertible matrices with integer entries

Let $\|\cdot \|$ be some matrix norm on the space of $n \times n$ matrices. Denote $$ M(r) := \{ A \in \mathrm{Mat}_{n \times n}(\mathbb{Z}) \mid \| M \| \leq r \}.$$ Denote by $p(r)$ the fraction of ...
Matthias Ludewig's user avatar
3 votes
1 answer
166 views

The spectral radius of a modified graph

Let $H$ be a graph and let $G=H \vee K_{1}$ be obtained by creating a new vertex and joining it to every vertex in $H$. This situation has many different names: $G$ is called the cone or the ...
Felix Goldberg's user avatar
3 votes
1 answer
309 views

Eigenvalues two-fold degenerate

Consider the matrix $$A:=\left( \begin{array}{cccc} 0 & a & 0 & 0 \\ f & 0 & b & 0 \\ 0 & e & 0 & c \\ 0 & 0 & d & 0 \\ \end{array} \right)$$ I ...
António Borges Santos's user avatar
3 votes
1 answer
135 views

On well separated circular regions in the Riemann sphere and complex polynomials

It started with a conjecture I had, see A statement on complex polynomials, which was false for $n \geq 3$, as shown by Noam D. Elkies in his answer there. The present post is an attempt to salvage ...
Malkoun's user avatar
  • 5,215
3 votes
1 answer
80 views

prove spectral equivalence bounds for fractional power of matrices

Let $A, D \in \mathbb{R}^{n \times n}$ be two symmetric,positive definite and tri-diagonal matrices for that we know that they are spectrally equivalent, thus ist holds $$ c^- x^\top D x \le x^\top A ...
Luna947's user avatar
  • 75
3 votes
1 answer
271 views

History of the characteristic matrix

Let $\mathbb{F}$ be a number field, $A$ and $B$ be two $n\times n$-matrix over $\mathbb{F}$. It is known from some textbook that $A$ is similar to $B$ iff there exists $n\times n$ nonsingular matrix $...
Zhihua Chang's user avatar
3 votes
1 answer
655 views

Upper bounds on the condition number of the eigenvector matrix

Let $A$ be an $n\times n$ real matrix with entries in a fixed interval $[a_\min,a_\max]$, with $a_\min$, $a_\max>0$. Question: Are there any upper bounds on the condition number of the ...
Ludwig's user avatar
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