**22**

votes

**0**answers

861 views

### Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices

In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...

**14**

votes

**0**answers

275 views

### a naive question about the second dual of a vector space

Let $K$ be a field. Are there non-scalar endomorphisms of the endofunctor
$$
V\mapsto V^{**}/V
$$
of the category of $K$-vector spaces?
I asked a related question on Mathematics Stackexchange, but ...

**14**

votes

**0**answers

361 views

### Cauchy matrices with elementary symmetric polynomials

$\newcommand{\vx}{\mathbf{x}}$
Let $e_k(\vx)$ denote the elementary symmetric polynomial, defined for $k=0,1,\ldots,n$ over a vector $\vx=(x_1,\ldots,x_n)$ by
\begin{equation*}
e_k(\vx) := \sum_{1 ...

**12**

votes

**0**answers

499 views

### How to explain the picturesque patterns in François Brunault's matrix?

How to explain the patterns in the matrix defined in François Brunault's
answer to the question Freeness of a Z[x] module depicted below? --
Choosing colors according to the highest power of 2 which ...

**11**

votes

**0**answers

561 views

### Regular languages of matrices and their generating functions

My question is somewhat related to this question.
Let us fix natural numbers $k$ and $C$. Let $A$ be an automaton whose alphabet consists of $k\times k$ matrices with integer coefficients of ...

**10**

votes

**0**answers

216 views

### Who stated and proved the “Hopf lemma” on bilinear maps?

If $A\otimes B\rightarrow C$ is a nondegenerate linear map, where $A, B, C$ are vector spaces over an algebraically closed field, then $\dim C\ge \dim A + \dim B -1$.
Nondegenerate here means ...

**10**

votes

**0**answers

271 views

### Matroids with prescribed independent sets

Let $A$ be a finite set. Let $B$ be a family of subsets of $A$. We are interested in a matroid with a minimum rank such that every element of $B$ is independent. The answer is obvious - a uniform ...

**9**

votes

**0**answers

377 views

### Which doubly stochastic matrices can be written as products of pairwise averaging matrices?

A matrix $A$ is called doubly stochastic if its entries are nonnegative, and if all of its rows and columns add up to $1$. A subset of doubly stochastic matrices is the set of pairwise averaging ...

**8**

votes

**0**answers

193 views

### Lift chain complex from $F_2$ to $Z$

We start with a finite dimensional chain complex over $F_2$, equipped with a basis. That is, we have finitely many finite dimensional $F_2$-vector spaces $C_0,\dots,C_k$ with bases $B_0,\dots,B_k$, ...

**8**

votes

**0**answers

234 views

### Eigenvalues of permutations of a real matrix: how complex can they be?

This is sort of complementary to this thread. I’ll repeat the definitions here:
For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and ...

**8**

votes

**0**answers

236 views

### An identity for Hankel determinants

Is the following result about Hankel determinants known or a simple consequence of some known results?
Let
$f(x) = \frac{\displaystyle 1}{{\displaystyle 1 - \frac{{a x^{m + 2}}}{\displaystyle {1 - ...

**8**

votes

**0**answers

434 views

### Bounding sum of first singular values squared for Kronecker sum of traceless matrices

Let $A$ and $B$ be $4\times4$ traceless matrices with Hilbert-Schmidt norms summing up to $1/4$, i.e.
$$\text{Tr}\left[ A\right]=\text{Tr}\left[ B\right] = 0,\qquad\text{Tr}\left[ A^\dagger A + ...

**8**

votes

**0**answers

498 views

### An elementary linear algebra problem

Let $K$ be a field, and let $E$ be the algebra of $n\times n$ matrices over $K$. Let $V_0$ and $V_1$ be the (left) $E$-modules of matrices of size $n\times n_0$ and $n\times n_1$. Let $W \subseteq ...

**8**

votes

**0**answers

196 views

### Generalized Classical Adjoints and Factorizations of the Characteristic Polynomial

This is idle noodling, and I'm prepared to learn that it's foolish as well as idle. But....
Let $M$ be an $n\times n$ matrix over, oh, let's say an algebraically closed field for now. There have ...

**8**

votes

**0**answers

396 views

### integral matrix of order p

Hi everyone
Let $p$ be a prime number. I am interested to classify $\{ A\in {\rm GL}_{p-1}(\mathbb{Z}): {\rm ord}(A)=p \}$ up to conjugacy. One reason to consider this problem is its relation to ...

**8**

votes

**0**answers

376 views

### On a tentative generalization of the Schmidt decomposition

Background
I am a PhD student in Physics and I am currently developing quite refined computer codes that allow to simulate many-body quantum systems living on a lattice. The difficulty resides in ...

**8**

votes

**0**answers

590 views

### Linear equations on unitary operators

Let $\alpha_1^{(s)},\dots, \alpha_n^{(s)} \in \mathbb{C}, \quad s=1, \ldots, r.$
Question:
Are the following conditions equivalent:
There are unitary operators $U_1, \ldots, U_n\in B(H)$ such
...

**8**

votes

**0**answers

614 views

### coordinate-free proof of transitivity of norms or traces

Hello:
Suppose $A$ is a finite free $B$-algebra and $B$ is a finite free $C$-algebra. Does anyone
know a coordinate-free proof (i.e. without choosing bases) of the identity:
$N_{A/C} = N_{B/C}\circ ...

**8**

votes

**0**answers

857 views

### roots of quadratic forms

This may be a very silly question, but I was wondering what is known about the roots of a quadratic form over variables $x_1,\ldots,x_n,y_1,\ldots,y_m$ in the finite field $\mathbb{F}_p$. I'm not ...

**7**

votes

**0**answers

84 views

### When is a product of hyperbolic matrices hyperbolic?

Suppose $A_1,\ldots,A_n$ is a sequence of $2 \times 2$ complex matrices such that $| \det(A_j) | =1$ and $ | \mathrm{tr}(A_j) | > 2 $ for each $j$. What kinds of reasonable restrictions can one ...

**7**

votes

**0**answers

179 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 ...

**7**

votes

**0**answers

115 views

### Problems where Conjugate gradient works much better than GMRES

I am interested in cases where Conjugate gradient works much better than GMRES method.
In general, CG is preferable choice in many cases of SPD because it requires less storage and theoretical bound ...

**6**

votes

**0**answers

120 views

### A variant of an Eventown problem for modulo a prime number

Consider the following problem, called the 'Eventown problem':
In a town, residents can form different clubs. The town council establishes the following rules:
1) Every club must have an even ...

**6**

votes

**0**answers

295 views

### On the number of polynomials that divide $x^{q-1}-1$ in some subspaces of $\mathbb F_q[x]$

Let $\mathbb F_q$ be a finite field (where $q$ is in general a power of a prime), and let $e, k$ be positive integers with $k \leq e < q-1$. Let $f_0(x), \ldots,
f_k(x) \in \mathbb F_q[x]$ be ...

**6**

votes

**0**answers

308 views

### Spaces of matrices with same eigenvalue/Great circles in O(n)-orbits

Let $Sym^2(V)$ be the set of symmetric matrices of a real $n$-dimensional vector space $V$. Given an element $\underline{\lambda}=[\lambda_1,\ldots \lambda_n]\in \mathbb{RP}^n$, where ...

**6**

votes

**0**answers

272 views

### maximal subgroups of $GL_2(Z/p^kZ)$

Hello,
is there any classification of proper maximal subroups of $GL_2(\mathbb{Z}/p^k\mathbb{Z})$ for $k>1$ (analogous to the one which exist for $GL_2(\mathbb{Z}/p\mathbb{Z})$)?
Could you give ...

**6**

votes

**0**answers

622 views

### Inverse of a matrix with binomial coefficients

Let $a(n,k)=(-1)^k {{2n-k}\choose k}$ for $0 \le k \le n$ and $a(n,k)=0$ else. Then it is known (cf. OEIS A005439 and A098435) that the first column of the inverse matrix of $(a(i,j))_{i,j\ge0}$ is ...

**6**

votes

**0**answers

337 views

### Can we write unitary matrices as positive linear combinations of Hermitian matrices?

The space $M_n:=M_n(\mathbb{C})$ of complex $n\times n$ matrices has the structure of a finite-dimensional complex vector space.
The space of Hermitian matrices forms a cone in this vector space ...

**6**

votes

**0**answers

208 views

### Is there a straightforward way to solve unmixed, homogeneous systems of polynomials?

I came across this problem in my research. It might just be an easy algebraic geometry question, but I don't know much algebraic geometry.
Suppose we have a system of $k\leq n$ polynomials in ...

**5**

votes

**0**answers

191 views

### Singularity of an $l\times l$ matrix whose entries are $2l$-th roots of unity

Let $l$ be a positive integer, $\zeta$ be a primitive $2l$-th root of unity in $\mathbb{C}$, and $\alpha,\beta$ be $\pm1$ sequences of length $l$, i.e. $\alpha_k=\pm1,\beta_k=\pm1$ for ...

**5**

votes

**0**answers

175 views

### Hermitian forms over quaternion algebra

Notations: Let $Q=(a,b)$ be a quaternion algebra over a field of characteristic $\neq 2$, i.e. $i^2=a, j^2=b, k=ij, ij=-ji$. Consider $K=k(t)(\alpha)$, where $\alpha=\sqrt{at^2+b}$. Let ...

**5**

votes

**0**answers

85 views

### Inverses of the sums of all possible subsets of a set of symmetric and positive definite matrices

I have a set of $c$ matrices $A_1 ... A_c$ which are all symmetric and positive definite. I would like to calculate the inverses of all the possible sums, i.e.
...

**5**

votes

**0**answers

223 views

### How to calculate the inverse of a matrix with block Kronecker structure?

I have a Hermitian matrix $A$ with the following block Kronecker structure,
$A =
\begin{pmatrix}
R_0 \bigotimes S_0 & R_1 \bigotimes S_1\\
R_1^H \bigotimes S_1^H & R_0 \bigotimes S_0
...

**5**

votes

**0**answers

159 views

### Largest entry of the inverse matrix?

I wonder if there is a "qualitative way" of predicting from the structure ix of the matrix $A$ which entry of $A^{-1}$ will be the largest. I am specially interested in the case
that $A$ is a ...

**5**

votes

**0**answers

131 views

### Numerical linear algebra: how to compute $B^TC^{−1}B$ efficiently

Hi,
my question is similar to this one. I have to compute $B^TC^{−1}B$, where $C$ is a strictly positive definite $n\times n$ matrix and $B$ is $n\times m$.
The matrix $C$ is huge ($n$ up to a ...

**5**

votes

**0**answers

320 views

### Determine if a matrix is unimodular

Is deciding if an integer square matrix has determinant $\pm 1$ faster that calculating the determinant of the matrix?

**5**

votes

**0**answers

402 views

### A reference on semisimple linear algebra

Is there any literature where the tools familiar from (multi)linear algebra are systematically transferred to the setting of semisimple modules over noncommutative rings?
In fact this question is a ...

**5**

votes

**0**answers

124 views

### reference for perturbation of projection result

Let $A$ and $B$ have the same rank and dimensions. If $P_A$ denotes the projection onto the range space of $A$, then
$$
\|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2).
$$
...

**5**

votes

**0**answers

188 views

### concentration for eigenvectors

I am interested in bounding from above the ratio between the maximum and minimum entries of a Perron vector. The only results that I found in the literature are from the classic masters (Ostrowski and ...

**5**

votes

**0**answers

490 views

### Status of Hadamard matrix conjecture

I would like to know if any progress has been made on Hadamard conjecture : There exists a Hadamard matrix of order $n=4k$ $\forall k \in \mathbb{N}$.

**5**

votes

**0**answers

347 views

### A class of determinants associated to Catalan-like Hankel determinants

The following matrices are related to some Catalan-like Hankel matrices. My question is whether direct computations of determinants of such matrices (i.e. without recourse to Hankel determinants) ...

**4**

votes

**0**answers

75 views

### Groups of operators between local unitaries and full unitaries

Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...

**4**

votes

**0**answers

202 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 $X,Y \in \mathbb{R}^{N\times d},d<N,\dim(X)=\dim(Y)=d$, ...

**4**

votes

**0**answers

84 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 ...

**4**

votes

**0**answers

184 views

### Probability distribution function for singular value sum of Gaussian random matrix

Let $\mathbf{X}$ be an $N \times N$ random matrix with IID Gaussian entries. They can be standard normal, but $N$ is not large: that is $N$ $<$ 6, typically. Call its singular value decomposition ...

**4**

votes

**0**answers

68 views

### Level sets of linear combinations of Gaussians

I am trying to work out whether level sets of linear combinations of Gaussian functions are unique.
For a given integer $n\ge 1$, fix $n$ points $x_i\in\mathbb{R}^d$ and $\sigma>0$. Let ...

**4**

votes

**0**answers

99 views

### Concept of eigenvector restricted to nonnegative entries

Let $X\in \mathbb{R}^{n\times n}$ be a positive semidefinite matrix. The leading eigenvector $v\in \mathbb{R}^n$ of $X$ is the solution to the problem
$\arg \max_{v:\lVert v\rVert_2=1} \lambda\quad$ ...

**4**

votes

**0**answers

94 views

### Preconditioner for finding the smallest eigenpairs of a large, but structured, matrix

I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $L$. $L$ is a graph Laplacian, with the following structure: $L = D ...

**4**

votes

**0**answers

103 views

### What is the composition in SesquiAlg?

To motivate my question, I will describe a famous 2-category. First, there is the 1-category $\text{Vect}$ of vector spaces (over some fixed field). This category has a tensor product $\otimes$, and ...

**4**

votes

**0**answers

191 views

### On the linear transformation between matrix space

Suppose we have two matrix subspaces, $n\times n$ matrix subspace $S_1$ and $m\times m$ matrix subspace $S_2$. Every element of $S_1$ and $S_2$ is complex symmetric matrix.
Suppose there exists ...