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

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4
votes
2answers
206 views

An algorithm to compare two representations of a simple Lie algebra?

I have two representations of a simple (complex or real) finite-dimensional Lie algebra $S$, both given in terms of their structure constants on a given basis. the first one is the adjoint ...
3
votes
1answer
208 views

Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$

I am looking for a explicit isomorphism between $Cl(8)$ (Clifford algebra over $\mathbb{R}^8$ with standard Euclidean metric) and $\mathbb{R}(16)$ (algebra of $16\times 16$ matrices over ...
2
votes
0answers
48 views

Computing maximal ideals of a Lie algebra

Would you know an algorithm (or an automatic method) that computes all maximal ideals $J$ of a given Lie algebra? Or an algorithm that computes all maximal ideals $J$ containing a given minimal ideal ...
5
votes
1answer
310 views

Examples of Polyhedra with Large Shadows

Let $P \subseteq \mathbb{R}^n$ be a polyhedron described by $\mathcal{O}(n^{c_1})$ inequalities, where $c_1$ is a constant. Moreover, let $M\colon P \to \mathbb{R}^2$ be a linear mapping. I'm looking ...
0
votes
1answer
80 views

What is exponentially fitted osculating straight line?

While reading an article about iterative methods for solving nonlinear equations I can't understand what is exponentially fitted osculating straight line. Could someone please briefly explain this ...
0
votes
1answer
92 views

When is a $2$-lift of a graph connected? [closed]

Let $\ (V\ E)\ $ be a graph, i.e. $\ E\subseteq\binom V2.\ $ A $2$-lift pattern of a graph is a function $\ e:E\rightarrow\{-1\,\ 1\}.\ $ The induced 2-lift is defined as the graph $\ V\times\{-1\,\ ...
1
vote
0answers
56 views

Lower bound on difference between polynomials at moderate distance

Fix $r > 0$ and $k, n \in \mathbb{N}$. Also consider a function $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$. Let $x_{1},\ldots, x_{n+1}$ be points chosen uniformly from $[-r,r]^{d}$. For $1 \leq i ...
2
votes
0answers
132 views

Is there a symmetric basis for $\mathbf{Q}(x,y)$?

Consider $\mathbf{Q}(x,y)$, the rational functions in $x$ and $y$, as a vector space over $\mathbf{Q}$. Let $\sigma$ be the map interchanging $x$ and $y$. Is there a basis for $\mathbf{Q}(x,y)$ ...
3
votes
2answers
169 views

Extremal eigenvalues & eigenvectors of skew-adjacency matrix

I am looking for ways to obtain the extremal eigenvalues and eigenvectors of the skew-adjacency matrix of a directed graph without diagonalizing it. The graphs I am interested in are not regular (but ...
0
votes
0answers
66 views

Signed Laplacians and Ramanujan graphs

Given a signing/2-lift matrix $A_s$ of a $d-$regular graph one has the relationship that the ``Signed Laplacian" is $L = d + A_s$. This $L$ is still the same size as the base graph. But the lifted ...
1
vote
1answer
59 views

Laplacian spectrum of $2-$lifts of graphs

We know that a $2-$ lift of a graph is specified by a $\pm 1$ assignment on the edges of the graph ( given as a signing matrix) denoting which edge is to be duplicated by the identity permutation on ...
3
votes
1answer
219 views

How to calculate the square root of matrix $A+B$ perturbatively?

$A=diag\{\lambda_1,...,\lambda_n\}$ and $\lambda_i>0$, $B$ is a positive definite symmetric matrix and $max\{B_{ij} \}\ll min\{\lambda_i\}$ Note that the perturbative calculation of square root ...
2
votes
2answers
151 views

Positive solutions of linear systems with a diagonally dominant matrix

Given a real linear system ($\mathbf{A}\mathbf{x} = \mathbf{b}$), is there any result regarding the positiveness of the solution $\mathbf{x}^*$ considering that $\mathbf{A}$ is diagonally dominant? ...
2
votes
0answers
107 views

Number of degree $k$ functions [closed]

Given a Boolean function $f:\{0,1\}^n\rightarrow\{0,1\}$, there is a real multivariate multilinear polynomial that is associated with in through interpolation. Example: ...
0
votes
2answers
90 views

Simple Spectrum of Jacobi matrices

I want to call a matrix a Jacobi matrix (cause there may be different notions of Jacobi matrices) if it is a tridiagonal matrix with positive off-diagonal entries. Now, I read that the spectrum of ...
0
votes
1answer
142 views

Are these particular kinds of matrices well known?

Given two positive integers $n$ and $a \leq \frac{n}{2}$ consider a $n \times n$ matrix $A$ such that, all the diagonal entries are either $a$ or $a+1$ all the non-zero off-diagonal entries are ...
0
votes
2answers
97 views

About adding a negative definite rank-1 matrix to a symmetric matrix

If $B$ is a symmetric matrix then how do its eigenvalues compare to the eigenvalues of $B - vv^T$? ( where $v$ is a vector of the same dimension as $B$) I guess that the eigenvalues of $B - vv^T$ ...
1
vote
0answers
37 views

Tools to bound the singular values of a finite sum of random matrices from below?

Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
5
votes
0answers
199 views

Existence of a matrix product from its eigenvalues

Let A and B be two positive definite, real, symmetric matrices. The eigenvalues of A, B and AB, denoted by $\lambda(X)$, obey the relation (from Bhatia): $$ \lambda^\downarrow(A) \cdot ...
11
votes
5answers
777 views

Does k(X) have a k-basis for every set X, without AC?

This question is inspired by Pace Nielsen's recent question Does a left basis imply a right basis, without AC?. For any field $k$, the field $k(x)$ of rational functions in one variable has an ...
1
vote
1answer
136 views

Are there good ways of relating a minor to the full determinant?

Say $A$ is a $(n-1)\times (n-1)$ matrix and we augment it by a $n^{th}$ row and a column and get a $n \times n$ matrix $B$. Is there a nice way to relate $det(B)$ and $det(A)$ and the added row and ...
-1
votes
1answer
84 views

How to show the square root function of a positive semidefinite matrix is differentiable? [closed]

How to show the square root function of a positive semidefinite matrix is differentiable? In this context PSD means symmetric PSD.
1
vote
0answers
68 views

Distribution of a signal covariance matrix

A common estimation problem in signal processing assumes the following signal model \begin{equation} \mathbf{r} = \sum_{i=1}^{Q}\alpha_i\mathbf{s}\left(w_i\right)+\mathbf{n} \end{equation} where ...
2
votes
0answers
75 views

Determinant of the sum of a psd (Kronecker) matrix and a diagonal matrix?

Let $K = K1 \otimes K2$ where $K1$ and $K2$ are positive semidefinite matrices. Let $W$ be a diagonal matrix with positive entries. (Everything is real-valued.) I want to calculate or bound $\det ...
0
votes
0answers
52 views

Shift functor and the origin of the linear decalage isomorphism

Let $(\mathbf{Vec}_\mathbb{Z}(\mathbb{K}),\otimes,\tau)$ be the symmetric monoidal category of $\mathbb{Z}$-graded $\mathbb{K}$-vector spaces, where $\otimes$ is the tensor product of ...
8
votes
0answers
118 views

Intersection of Springer fibre and Schubert cell

Let us consider intersections of Springer fibres and Schubert cells in type A. Let $ Y : \mathbb C^n \rightarrow \mathbb C^n $ be a nilpotent operator. Let $$ F_Y = \{ V_0 = 0 \subset V_1 \subset ...
0
votes
0answers
90 views

Question about majorization of eigenvalues after conjugation

Let $A$ and $B$ be $n \times n$ positive semidefinite matrices with eigenvalues $\alpha_1 \ge \alpha_2 \ge \ldots \ge \alpha_n$ and $\beta_1 \ge \beta_2 \ge \ldots \ge \beta_n$ respectively. ...
0
votes
0answers
75 views

The norm of a Finite Hilbert matrix

Let $H$ be an $n\times n$ Hilbert matrix, $$h_{ij}=(i+j-1)^{-1}.$$ The matrix $p$-norm corresponding to the p-norm for vectors is: $\left \| A \right \| _p = \sup \limits _{x \ne 0} \frac{\left ...
12
votes
3answers
864 views

How do we show this matrix has full rank?

I met with the following difficulty reading the paper Li, Rong Xiu "The properties of a matrix order column" (1988): Define the matrix $A=(a_{jk})_{n\times n}$, where $$a_{jk}=\begin{cases} ...
1
vote
0answers
126 views

Distinct determinants of circulants

If $M$ is a circulant integer matrix of size $n\times n$ whose entries are randomly chosen from $\{0,1\}$ value, how many different determinants does $M$ possibly take value in? For $n=1,2,3,4$, I ...
3
votes
0answers
99 views

Dimension of certain polynomial spaces

Let $(\omega_1, \eta_1) \dots (\omega_n, \eta_n)$ be $n$ pairs of complex numbers where $\omega_i \ne \omega_j$ for all $1 \leq i \ne j \leq n$. We define the following polynomial space $$ Z_n^d(\eta, ...
3
votes
1answer
310 views

Open problems in compressed sensing

What are the main open problems in compressed sensing? I am interested in theoretical as well as in numerical point of view.
4
votes
2answers
224 views

Eigenvectors of a symmetric positive definite Toeplitz matrix

I wish to efficiently compute the eigenvectors of an n x n symmetric positive definite Toeplitz matrix K. A full eigendecomposition would be even better. Although I assumed this would be a well ...
1
vote
0answers
174 views

On matrix rank inequality

Let $A$ be a $\{0,1\}$ square matrix. Let $J$ be all $1$ matrix. Let $\bar{A}=J-A$. Is it possible for $rk_+(A)\geq c\cdot rk_+(\bar{A})^d-1$ and $rk_+(\bar{A})\geq c\cdot rk_+({A})^d-1$ for some ...
6
votes
1answer
200 views

Diagonalization of the matrix $(1/(i+j+\rm{const}))_{i,j}$

Consider the following infinite matrix: $A_{i,j}=\frac1{i+j+\gamma}$, $0\leq i,j<\infty$, $\gamma>0$ is a constant. Is it known how to diagonalize $A$, or, say, calculate $(I+tA)^{-1}$ for ...
6
votes
1answer
168 views

Exotic “non-linear” (but “almost linear”) automorphisms of symplectic vector space

Let $V$ be a vector space over a field $k$ equipped with a symplectic form $\omega$. Let $f:V \rightarrow V$ be a bijective set map such that the following hold. For all $v \in V$ and $c \in k$, we ...
0
votes
0answers
43 views

Canonical forms of symmetric/skewsymmetric quaternionic matrix

$A$ belongs to $n$-dimensional quaternion symmetric matrix, in the sense that $A=A^T$, where $T$ means transpose. Under transformation $U$, $A\rightarrow U\cdot A\cdot U^T$, where $U$ is $n$-dim ...
1
vote
0answers
30 views

Inverse of the covariance of the estimate of a covariance

I have a covariance matrix, $V_{ij}$, which (for reasons that aren't important) I'm going to call the visibilities. I have an estimator for the visibilities $\hat V_{ij}$, and I've derived that the ...
1
vote
1answer
151 views

finding permutation matrix I which minimizes TRACE( I* C*( I^T)* D) matrix

I have a problem that is really important for my thesis and i am not studding math so i will be very glad if you help me in this case... thanks for your help in advance I want to find permutation ...
0
votes
0answers
132 views

$Ax=b$ in a function space (again)

Let $X$ be compact Hausdorff topological space, $C(X)$ denote the algebra of complex-valued continuous functions on $X$, $b\in \mathbb{C}^m$, $\mathbf{A}\in C(X)^{m\times n}$, Let ${\mathbb{C}}^n$ ...
0
votes
2answers
81 views

Tensor product-definition-balanced versus bilinear maps

When defining tensor products $M\otimes_R N$ over a commutative ring $R$ one can use a universal property with respect to bilinear maps $M\times N\rightarrow P$, for any $R$-module $P$. On the other ...
0
votes
3answers
163 views

Infinite dimensional vector spaces

This question might seem elementary but I cannot answer it. Let M be an infinite dimensional vector space and $f_1, f_2, \cdots , f_r \in M^*$ be a set of linear independent vectors with $r \geq 2$. ...
1
vote
0answers
102 views

How to prove the following claim about Pseudoinverse

For real symmetric matrices $K$ and $\hat{K}$, if $u^{T}{K}u\le u^{T}\hat{K}u\le C u^{T}{K}u$ for all $u$ in the row space of $K$, then $u^{T}{K^{+}}u\ge u^{T}\hat{K}^{+}u\ge u^{T}{K^{+}}u/C$ for all ...
8
votes
3answers
623 views

Does a left basis imply a right basis, without AC?

If $_DV_D$ is a $D$-$D$-bimodule, and we have a $D$-basis for $V_D$, do we still need AC to get a $D$-basis for $_DV$? (The original question appears below. But this shorter question gets at the ...
2
votes
1answer
73 views

Bringing a (Least Squares Problem) Matrix into Block Upper-triangular Shape via Matrix-reordering

I have the problem of solving very large and very sparse least squares problems and, a bit dissatisfied with the run-times of the full-fledged QR-algorithm, I would like to bring the instances into ...
7
votes
1answer
148 views

Is there a ring which is not Hermite but is coherent?

Call a commutative unital ring $R$ Hermite if for all $m, n\in \mathbb{N}$ with $m<n$, and all $f\in R^{m\times n}$ such that transpose($f$) is left invertible (with a matrix with entries from ...
5
votes
2answers
202 views

Compute adjugate matrix over commutative ring

Let $A$ be a $n\times n$ matrix over a commutative ring. I'm looking for a good method to compute its adjugate matrix. My current approach is to use the Cayley-Hamilton theorem: $$\text{adj}(A) = ...
1
vote
1answer
112 views

Linear map of Zonotopes [closed]

Consider a linear system with a map $A: \mathbb{R}^n \rightarrow \mathbb{R}^m$ such that $y = A x$ with $n \geq m$ The input space $x$ is constrained by a zonotope set $\mathcal{X} \subseteq ...
0
votes
1answer
151 views

$Ax=b$ in a function space

Let $X$ be compact Hausdorff topological space, $C(X)$ denote the algebra of complex-valued continuous functions on $X$, $b\in \mathbb{C}^m$, $\mathbf{A}\in C(X)^{m\times n}$, for all $x\in X$, ...
3
votes
2answers
269 views

(linear algebra) - Can a symmetric equilibrium achive higher social-welfare than some equilibrium with the same support?

EDIT: rewritting the question to linear algebra to make it more accessible. Denote by $\Delta([n])$ the set of all probability distributions over $\{1,2,\ldots,n\}$, that is: ...