# Tagged Questions

**1**

vote

**0**answers

12 views

### Wedge product of entries of a matrix & Volume form of the Siegel metric

Let $A=(a_{ij})$ be an $n\times n$ square matrix, and $\omega(A)=\bigwedge\limits_{i,j=1}^na_{ij}$ be the wedge product of its entries. Then, if $B=UA=(b_{ij})$ for some square matrix $U$, I think one ...

**-2**

votes

**0**answers

42 views

### proof Hadamard's Inequality [on hold]

Theorem 4.2. Hadamard’s Inequality. Suppose A is positive semideﬁnite
of size n. Then |A|≤ [A]11···[A]nn.
Proof. Let A be any positive semideﬁnite matrix of size n. Note that
In is a postive ...

**-1**

votes

**0**answers

27 views

### Determinants of block matrices with non-square diagonal and square anti diagonal elements [on hold]

Is there a way to find the determinant of $X$ in terms of its sub-matrices $A,B,C_0$ and $R_0$?
$$X = \begin{bmatrix}
AC_0 & -I_n\\
0_{(n-1)} & R_0B
\end{bmatrix} \in \mathbb{R}^{(2n-1) \...

**0**

votes

**0**answers

51 views

### conjugacy classes of cyclic subgroups of order $k$ in $ {\rm GL}_n(\mathbb{Z}/p^m\mathbb{Z}) $

Let $p$ a prime numbers and $k$ be positive integer such that $(k, p) = 1$. And $m$ be the order of $p$ in $\mathbb{Z}/k\mathbb{Z}^×$. How many conjugacy classes of cyclic subgroups of order $k$ does
...

**0**

votes

**0**answers

38 views

### Why is $R_{\Omega}R_{\Omega}^{*}=R_{\Omega}$ in matrix completion?

In the matrix completion literature, a restriction or sampling operator is defined as follows. Let $X$ be the unknown matrix to be recovered and $\{w_{\alpha}\}_{\alpha=1}^{n^2}$ be some orthonormal ...

**5**

votes

**1**answer

110 views

### Can a projective solvable group be transitive?

Let $p > 3$ be a prime number, and let $G \leq \mathrm{PGL}_2(\mathbb{F}_p)$ be a solvable subgroup.
Is it possible that the action of $G$ on $\mathbb{P}^1(\mathbb{F}_p)$ is transitive?

**1**

vote

**0**answers

72 views

### Number of conjugacy classes of cyclic subgroups of order $pq$ of $GL_n(\mathbb{Z}/p\mathbb{Z}) \times GL_m(\mathbb{Z}/q\mathbb{Z})$

Let $p$ and $q$ be distinct prime numbers, and let $m$ and $n$ be positive
integers. How many conjugacy classes of cyclic subgroups of order $pq$ does
the group
$$
{\rm GL}_n(\mathbb{Z}/p\mathbb{Z})...

**4**

votes

**0**answers

92 views

### Whether a given algebra is the algebra of endomorphisms for a vector space

Let $\mathbb{F}$ be a field and let $A$ be an associative unital $\mathbb{F}$-algebra. Is there a criterion to let me know if $A$ is isomorphic to the algebra $\mbox{End}(\mathbf{V})$ of endomorphisms ...

**2**

votes

**2**answers

186 views

### Need help with paper written in Russian… Yorgov's paper on self-dual codes with automorphisms of odd order

I came across this paper by V.I.Yorgov named "Binary self-dual codes with automorphisms of odd order". (Actually I was first reading another paper by Borello that cited this paper. It later become ...

**5**

votes

**1**answer

76 views

### Positive semidefinite ordering for covariance matrices

Suppose that X and Z are matrices with the same number of rows. Let
$$ D = \left[\begin{array}{cc} X' X & X'Z \\ Z'X & Z'Z \end{array} \right]^{-1} - \left[\begin{array}{cc} (X' X)^{-1} & ...

**-1**

votes

**0**answers

29 views

### Relation between Independent variables in an Equation [on hold]

Description: We define index as an indicator, sign, or measure of something.
Let, $A_{i}$ is an index, that measures the benefits of choosing a network station $i$ among other existing network ...

**1**

vote

**1**answer

103 views

### Cluster algebra structure on the coordinate ring of $Mat_3$

Let $Mat_3$ be the set of all 3 by 3 matrices. I have some questions on the cluster algebra structure on the coordinate ring of $Mat_3$.
We use $\Delta_{j_1\ldots j_n}^{i_1\ldots i_n}$ to denote the ...

**0**

votes

**1**answer

70 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 ...

**0**

votes

**0**answers

94 views

### Sum of unit vectors always has a binary span after constrained permutations

Conjecture:
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, ...

**3**

votes

**1**answer

122 views

+100

### 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{...

**6**

votes

**0**answers

72 views

+100

### 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

48 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}...

**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\...

**2**

votes

**1**answer

78 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.
...

**0**

votes

**0**answers

62 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 ...

**1**

vote

**1**answer

30 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 ...

**5**

votes

**1**answer

235 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 ...

**1**

vote

**1**answer

159 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}&...

**0**

votes

**1**answer

84 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

209 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 \{...

**5**

votes

**1**answer

81 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 ...

**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

785 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 ...

**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. ...

**1**

vote

**1**answer

184 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 $r \times n$ ...

**-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]
...

**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 ...

**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 ...

**3**

votes

**3**answers

576 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 ...

**3**

votes

**1**answer

66 views

### How does grade projection act on homogeneous multivectors in geometric algebra?

I'm reading Clifford Algebra to Geometric Calculus by Hestenes, and struggling with an early result about reversion inside of a grade-projection operator.
It is noted that $A_r$ and $B_s$ are ...

**0**

votes

**0**answers

55 views

### Image of composition of integral upper triangular matrices

For $A,B$ integral upper triangular matrices on $\mathbb{Z}^k$, do we know something about the image $\text{im}(AB)$ in terms of $\text{im}(A)$, $\text{im}(B)$, unions, intersections, determinants, ...

**1**

vote

**0**answers

104 views

### Generating $\mathfrak{so}(7)$

Short version: Let $V$ be a 7-dimensional linear space of (real) square matrices. Suppose further that $[V,V]$ (the linear space spanned $[X,Y]$, $X,Y\in V$) is actually a subalgebra isomorphic to $\...

**0**

votes

**0**answers

62 views

### Checking whether a given matrix has a non-zero determinant

For a positive integer $n$, let $c$ be the number of ordered integers tripartitions $(a_j,b_j,c_j)$ of $n$.
Now consider the $c \times c$ matrix $M$ in which the value of the $M[i,j]$ is
$M[i,j]={(...

**5**

votes

**1**answer

184 views

### Bijection modeling isomorphism of infinite-dimensional vector spaces

Let $T : V \to W$ be an isomorphism of vector spaces with bases $B_V$ and $B_W$, which may be of any cardinality.
Does there exist a bijection $f : B_V \to B_W$ such that, for each
$b_V \in B_V$,...

**2**

votes

**0**answers

60 views

### How to find a closed form of following matrix's determinant [closed]

I wanna find a closed form of determinant of the following matrix
$$A(n) =
\begin{pmatrix}
B_{1} & B_{2} & \cdots & B_{n} & 1 \\
B_{n} & B_{1} & \cdots & B_{n-1} &...

**0**

votes

**0**answers

45 views

### Variant of Holder's inequality [migrated]

So far I believed that only the reverse Holder inequality holds for $0<p<r<1,$ but then a student pointed out to me that
$$\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$$ A few numerical ...

**1**

vote

**0**answers

41 views

### Basis for a set of polynomials in Sage? [closed]

I have a large set of polynomials in the coordinates $x,y,z$ in Sage, (e.g. $x^5y-3x^2y^2+2xy^3+x^2yz-y^2z$). I want to know, for example, if $x^5y$ is in the span of my set. Is there a Sage command ...

**3**

votes

**1**answer

143 views

### The spectral norm of the truncated exponential of a matrix

Let $A$ be a matrix satisfying $A^*+A\leq0$, it can be shown that $\|e^{tA}\|_2\leq1$ for all $t\geq 0$, where $\|\cdot\|_2$ is the spectral norm defined as largest singular value of the matrix.
I am ...

**3**

votes

**1**answer

330 views

### What is the mathematics behind the random experiment which produces the data with this strange property?

I have a following scenario. there is a huge collection of data resulting from a random experiment $E$ (I do not say random variable yet, for reasons that you will need to explain in your answer). Let ...

**0**

votes

**1**answer

83 views

### Construction of orthonormal basis of the Hilbert space $\mathcal{S}^p_{\mathcal{H}}$ of vectors of $p \in \mathbb{N}$ Hilbert Schmidt operators

Let $(e_j)$ be a orthonormal basis (ONB) of a separable Hilbert space $(\mathcal{H}, \langle\cdot, \cdot\rangle_{\mathcal{H}})$ and $(\mathcal{S_H}, \langle\cdot, \cdot\rangle_{\mathcal{S_H}})$ be the ...

**1**

vote

**0**answers

47 views

### Determinant formula related to solutions of a second-order recurrence

Let $A$ be the linear map on the space of complex sequences acting as
$$(Au)_{n}=u_{n-1}+a_{n}u_{n}+u_{n+1}, \quad n\in\mathbb{Z},$$
where $\{a_{n}\}$ is a fixed sequence. Let $f=f(z)$ and $g=g(z)$ be ...

**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 ...