Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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2
votes
2answers
112 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
1answer
47 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
0answers
27 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
1answer
98 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
1answer
68 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
0answers
76 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
1answer
88 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
0answers
49 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
0answers
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}...
-3
votes
0answers
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
0answers
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
0answers
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
0answers
56 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
votes
0answers
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 $...
2
votes
1answer
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. ...
0
votes
0answers
61 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
1answer
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
1answer
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
4answers
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
1answer
234 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
1answer
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}&...
0
votes
1answer
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
1answer
203 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
1answer
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 ...
1
vote
0answers
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
2answers
780 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
0answers
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
1answer
182 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
1answer
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
2answers
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
0answers
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
3answers
571 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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
183 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
0answers
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
0answers
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
0answers
40 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
1answer
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
1answer
329 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
1answer
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
0answers
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
1answer
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 ...
4
votes
1answer
81 views

information measure for matrix that is analogous to rank

Is there a measure for matrix that is analogous to rank of the matrix, but it is continuous on matrix elements? Say, we could say the information in identity matrix $I_n$ is $n$, and when the off-...
2
votes
0answers
33 views

Why does the objectivity rule out the convexity?

In the famous work "Ball J M. Convexity conditions and existence theorems in nonlinear elasticity[J]. Archive for rational mechanics and Analysis, 1976, 63(4): 337-403", it was mentioned that the ...
1
vote
0answers
31 views

Practical application of envelope theorem for linear programs

Assume that we have solved a (standard) linear program $$ \text{minimize}_{x\in {\mathbb R^n}}\,\, c_0^Tx, \,\,\,\,\, \text{s.t. } A_0x \leq b_0, $$ and would like to know how sensitive is the optimal ...
0
votes
0answers
27 views

homogeneous singular pencil of matrices

I was reading singular pencil of matrices from F.R Gantmachers book(Vol 2, Chap 12), where he deals with the strict equivalence of two pencil, to introduce the concept of infinite elementary divisor ...
0
votes
0answers
85 views

On the transitivity of the action of the unitary group

Let $H$ be a complete inner product space over either real or complex numbers. If $H$ is complete, for two finite sets of vectors $\left\{e_i\right\}_{i\in I}$ and $\left\{f_i\right\}_{i\in I}$ there ...