**2**

votes

**0**answers

47 views

### A specific spanning property of a family of vectors

Let $v_1, v_2, v_3, \dots$ be a family of vectors in $\mathbb R^n$ that span $\mathbb R^n$. Given this family, define the family of vectors
\begin{align*}
\begin{pmatrix} v_1 \\ \alpha_1 v_1 \end{...

**0**

votes

**1**answer

49 views

### Non-strict column diagonally dominant matrix inner product

Let $A \in \mathbb{R}^{n \times n}$ be a normalized non-strict column diagonally dominant matrix, that is:
$$a_{j,j} = \sum_{i \ne j} \left|a_{i,j}\right|$$
where $0 \le a_{j,j} \le 1$ and $-1 \le ...

**16**

votes

**1**answer

571 views

### Number of matrices with given Smith normal form

Denote with $\mathcal{M}$ the set of $(m \times n)$-matrices with integer coefficients bounded by some $K$. Given a matrix $B \in \mathcal{M}$ that is in Smith normal form, is anything known about the ...

**0**

votes

**0**answers

39 views

### Sums of unit vectors has a binary span after constrained permutations

Let $e_1 = (1,0,\ldots,0), \ldots , e_{m_1+m_2} = (0,\ldots,0,1)$ be the unit vectors of the standard basis $E$ of $\mathbb{R}^{m_1+m_2}$.
An enumeration $ E \cup -E = \{f_1, \ldots, f_{2(m_1+m_2)}\}...

**2**

votes

**1**answer

283 views

### integral basis of orthogonal complement

Suppose there are $r$ linearly independent vectors $v_1,\dots,v_r\in \mathbb{R}^n$, all of them have integer-valued entries and $\|v_i\|_\infty\leq m$ for some integer $m$.
My goal is to find an ...

**2**

votes

**0**answers

27 views

### Matrix semigroups in which a weighted average of eigenvalues is multiplicative

A problem in fractal geometry requires me to consider matrix semigroups with the following curious property. For a $d \times d$ real matrix $A$ let $\lambda_1(A) \geq \lambda_2(A) \geq \cdots \geq \...

**1**

vote

**0**answers

45 views

### Linear independence of an odd set of measurable functions

Let $g(t)$ be a convex positive function. I'm trying to show that the set $\{ |t|^ne^{\frac{g(t)}{t^n}}\}_{n\in \mathbb{N}}$ is linearly independent in the space of measurable functions on $\mathbb{R}...

**1**

vote

**1**answer

465 views

### Find the following transformation $G$

I asked this question 6 days ago on math.stackexchange.com (http://math.stackexchange.com/questions/656585/find-the-following-transformation-g). I didn't get any answers yet, so I'm posting here.
I'm ...

**0**

votes

**2**answers

499 views

### Independence in mathematics

While trying to think about possible interesting notions of algebraic independance over a skew field, I am wondering where in mathematics appears the notion of being independent, or free over ...

**3**

votes

**3**answers

566 views

### A question on linear groups

Asking this question I have made a mistake joining my main question with two simple ones, so it hasn't received enough attention, however there was a partial answer, which was not elaborated, and now ...

**2**

votes

**1**answer

75 views

### Minimize matrix distance to tensor product

Minimize the following function:
$ f(V) = || V \otimes V - U_1 \otimes U_2 ||$
where $U_1, U_2 \in SU(n)$ are fixed and we minimize over all $V \in SU(n)$. The norm is from the trace inner product.
...

**-3**

votes

**0**answers

62 views

### Determinant of a tensor product [closed]

Let V and W be two vector spaces over a field of characteristic zero.
Give a formula for the top exterior power of V tensor W.

**0**

votes

**0**answers

13 views

### Does a vector belongs to a simplicial subcone when it belong to cone with more than n generators?

Assume $x_{0}\in \text{cone}(a_{1},\dots,a_{N})$, where $a_{i}\in \mathbb{R}^{n}_{+}$ ($a_{i}\in \mathbb{R}^{n}$, and $a_{i}\geq 0$) for $i=1,\dots,N$ (i.e., $x_0$ lies in the cone generated by $a_{i}$...

**2**

votes

**0**answers

33 views

### Partially permutative matrices

Let $V$ be a finite dimensional vector space over a field $K$. Then a map
$L:V\otimes V\rightarrow V\otimes V$ is said to satisfy the Yang-Baxter equation if $(L\otimes I)(I\otimes L)(L\otimes I)=(I\...

**0**

votes

**0**answers

55 views

### An idea of two sided invertible matrix [closed]

Can you suggest me a reading material that expand on the idea of two matrices that are inverse to each other but which are not square matrices?
Here's what I think of, take $A$ a matrix of order $n\...

**1**

vote

**1**answer

42 views

### Approximate largest eigenvalue of Monodromy matrix

Does anyone know the procedure (or have pseudo code) to approximating the largest eigenvalue of a monodromy matrix? Or even to approximate the monodromy matrix itself?
There is no explicit solution ...

**-1**

votes

**0**answers

22 views

### Equality of sum of fractions implies correspondence of terms [closed]

I am working in a theorem of Jhonson and Newman about cospectrality and got stucked un this claim. can you help me?
$a_i$ and $b_i$ are non negative numbers, $z\in\mathbb{C}$ and $d_i \neq d_j$ for $...

**1**

vote

**1**answer

102 views

### $0/1$ programming multiple quadratic constraints

If we have an $n$-variable rank $n$-linear system it is clear we can find whether there exists a $0/1$ solution in polynomial time.
If we have an $n$-variable degree $2$ system how many constraints ...

**3**

votes

**1**answer

361 views

### Simultaneous action of GL(n) on matrices

Consider the action of $GL(n,k)$ on the set $M\times M$ where $M$ is the set of all $n$-by-$n$ matrices over $k$ given by $g\cdot(h,l) \mapsto (ghg^{-1}, glg^t)$.
Individually these actions are well-...

**0**

votes

**0**answers

60 views

### A matrix representation problem over finite field

Given a matrix $H\in\Bbb F_p^{n\times n}$ and a list of matrices $A_i,B_i\in\Bbb F_p^{n\times n}$ at each $i\in\{1,\dots,m\}$ where $m\leq n^{1+\beta}$ for some $\beta\in[0,1)$ with $\|H\|_0>n^{1+\...

**2**

votes

**1**answer

102 views

### Maximize inner product of a tensor of unitary matrices

How can one maximize the following function:
$ f(V) = || V \otimes V - U_1 \otimes U_2 ||$
where $U_1, U_2 \in SU(n)$ are given and we seek to maximize over $V \in SU(n)$.
Both the maximum value of ...

**2**

votes

**1**answer

139 views

### Possible lower bound in quantum many body system with non-local terms

I am asking a question related to Lieb-Robinson bound and nonlocality.
As we know from Lieb-Robinson theorem (see e.g. http://arxiv.org/abs/1008.5137): Suppose a Hamiltonian system is local, i.e. $H=\...

**3**

votes

**1**answer

167 views

### Minimal polynomial of unipotents in orthogonal group

Consider split orthogonal group $O(2l)$ over a field of characteristic zero. We may assume the matrix of the bilinear form to be $\begin{pmatrix} O&I\\ I&O\end{pmatrix}$.
Let $u$ be a ...

**1**

vote

**1**answer

157 views

### Is this a full rank matrix? [closed]

According to the answer of znt to the previous version, I revise the question as follows:
Is there a real $(n-1)\times n$ matrix $A$
such that $A$ is not a full rank matrix and satisfy $a_{ii}&...

**5**

votes

**1**answer

233 views

### How often does a pair of linear maps generate a Zariski-dense subgroup of $GL(d,\mathbb{R})$?

I am an analyst working on a number of problems which in some way relate to random matrix products. In this context I frequently find that the analytic properties I am interested in depend in some way ...

**14**

votes

**4**answers

700 views

### Groups of matrices in which all elements have all eigenvalues equal in modulus

I am writing a research article in which I need to use the following fact: if $G$ is a subgroup of $GL_3(\mathbb{R})$ which is irreducible in the sense that no proper nontrivial subspace of $\mathbb{R}...

**1**

vote

**1**answer

29 views

### nonnegative solution of nonhomogeneous under-determined linear system of equations

For a set of under-determined linear equations, I was wondering if there is any closed form for all non-negative solutions? Is there a way to analytically characterize the feasibility set of such ...

**19**

votes

**4**answers

1k views

### Nuances Regarding Naturality

It's frequently said, informally, that a natural isomorphism is one that doesn't depend on arbitrary choices.
But the phrase "arbitrary choices" lends itself to different interpretations. Consider ...

**4**

votes

**1**answer

243 views

### minimal polynomial for a graph

I wonder if there is any result relating the degree $d$ of the minimal polynomial of a directed finite graph to any of its topological features - such as its diameter, or any other similar 'natural' ...

**5**

votes

**1**answer

80 views

### Complexity for solving linear equations?

What is the best known complexity for finding a vector $x \in \mathbb{R}^n$ to minimize $||Ax - b||^2$ and/or to solve (when possible) the system of linear equations $Ax=b$?
I am interested in ...

**5**

votes

**2**answers

2k views

### What is the time complexity of truncated SVD

Full SVD, on an m*n matrix $A$, $[U,S,V] = svd(A)$, would cost $O(m^2n + mn^2 + n^3)$ time.
But what is the time complexity if we only need the $k$ largest singular values, say, $[U_k,S_k,V_k] = svds(...

**2**

votes

**0**answers

63 views

### How to find moment condition for generalized method of moments?

Consider a scalar system with $2K$ outputs and $K+2$ unknowns:
$y_{k,1}=x_ka_1+n_{k,1} \quad y_{k,2}=x_ka_2+n_{k,1}$.
The variables $n_{k,\ell}$ are zero mean noise variables.
To estimate $a_1$ and $...

**0**

votes

**1**answer

81 views

### Partitioning an orthogonal matrix into full rank square submatrices

Let $U$ be an $n \times n$ orthogonal matrix. Given an arbitrary partition ${\mathcal P}_c=\{y_1,y_2,\ldots,y_k\}$ of the columns of
$U$, does there always exist a corresponding partition ${\mathcal ...

**1**

vote

**1**answer

198 views

### Approximation of sets

Is the following true? For every $\varepsilon>0$ there is a finite
subset $W$ of $\mathbb{N}\times \mathbb{N}\times \mathbb{N}$, such that
$$|p_1(W)\cap p_2(W)\cap \{p_1(x)+p_2(x):x\in W\}\cap \{...

**1**

vote

**0**answers

46 views

### Expected amount of linearly dependent random vectors? [closed]

Given a random Matrix $A\in \mathbb{F}_2^{n\times n}$ what is the expectation value of the amount of linearly dependent row-vectors of $A$?
EDIT: As said in the comments, I'm looking for the ...

**12**

votes

**2**answers

776 views

### How are eigenvalues and eigenvectors affected by adding the all-ones matrix?

Given an $n \times n$ matrix $A$ and the $n\times n$ all-ones matrix $J = (1)_{ij}$, I'm interested in the relation between the eigenvalues and eigenvectors of the matrices $A$ and $A+J$, or more ...

**8**

votes

**1**answer

262 views

### $k[x_1, \dots, x_n]$ is free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism?

For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set $\...

**1**

vote

**1**answer

176 views

### An extremal problem on matrices

Is it possible to determine (or give bounds for) the following extremal problem:
Let $k,m,r$ be positive integers such that $k,m \geq r$. What is the least number $n$ such that for any $r \times n$ ...

**0**

votes

**0**answers

22 views

### showing that a matrix has repetitive values?

Here my primary aim is to calculate the stationary distribution of a DTMC using left-eigen values i.e, $ \pi = \pi*P$.
But for some matrices, I observe that some states a same stationary probability. ...

**-2**

votes

**1**answer

33 views

### Rotating a known vector over two axis-es to result to another known vector [closed]

Lets assume i have a known vector, for example x = [1,0,0]
After 2 rotations, one over the y axis and one over the z axis, i result in a vector which in this example is x' = [0.5774, 0.5774, 0.5774]
...

**5**

votes

**1**answer

310 views

### Bounded domain matrix factorization

Assume we're trying to find $A\in [-1,1]^{n\times d}, B\in [-1,1]^{m\times d}$, from an observed matrix $C\in [-d,d]^{n\times m}$, where $C=AB^T$.
The goal is to return $\widehat A, \widehat B$ such ...

**24**

votes

**2**answers

2k views

### Linear algebra in terms of abstract nonsense?

The categories of vector spaces and finite dimensional vector spaces are pretty much as nice as can be, I think.
I was wondering what portions of basic linear algebra (first couple of courses) fall ...

**12**

votes

**2**answers

994 views

### Jacobi's equality between complementary minors of inverse matrices

What's a quick way to prove the following fact about minors of an invertible matrix $A$ and its inverse?
Let $A[I,J]$ denote the submatrix of an $n \times n$ matrix $A$ obtained by keeping only the ...

**1**

vote

**0**answers

30 views

### distance from the mean of a normal distribution to the span of a random sample

Let $W$ be a $d\times k$ matrix whose columns are sampled from a multivariate normal distribution with mean $\mu$ and unit covariance. I'm interested in $|\mu - WW^+\mu|$, that is the distance from ...

**2**

votes

**0**answers

90 views

### When is an selfadjoint operatorvalued matrix with positive semidefinite diagonal elements positive semidefinite as well?

We have $p \in \mathbb{N}$ and $\mathcal{H}$ is a Hilbert space.
let's consider a matrix $\boldsymbol{\Gamma}_p := (C_{i-j})_{i,j=1, ..., p} \in \mathcal{S_H}^{p\times p}\!\!\,,$ that is a $p\times ...

**2**

votes

**1**answer

237 views

### Sampling from random totally unimodular matrices of a particular type?

Is there a way to parametrize totally unimodular $(3n+2)\times(2n+2)$ matrices of form
$$\begin{bmatrix}
\pm1 & \pm1 & 0 & 0 &\dots & 0 & 0 & 0 & 0\\
A_{2n} & B_{2n}...

**4**

votes

**1**answer

134 views

### Existence of Nonnegative Solutions of Linear Systems of Equations and Inequalities with particular constraints

Suppose we have an $n \times m$ nonnegative matrix $A$, where each row sums to $1$. I wonder whether there exists an $m \times n$ nonnegative matrix $X$ that satisfies the following constraints:
...

**3**

votes

**2**answers

255 views

### Rank of a fat random matrix

Let $\mathbf{R} \in \mathbb{C}^{~n \times k} $ with $n \leq k $ be a random matrix, whose entries are i.i.d zero mean random variables with circularly symmetric Normal distribution. Two questions:
(...

**1**

vote

**8**answers

964 views

### Bivectors in 3 and 4 dimensions

The big questions behind are:
Is a bivector a two-form?
Why a bivector is simply a vector in 3 dimensions?
How to distinguish between vectors and bivectors in 3D?
Why all bivectors are not vectors ...

**1**

vote

**1**answer

66 views

### Inverting two paraboloid relations

Let's suppose we have two variables $(\alpha, \beta)$ on two functions $k_1, k_2$ that can be defined in terms of matrix relations:
$$
k_1 = \left| \xi^T \mathcal{F}^T \mathcal{F} \xi \right|
$$
$$
...