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

learn more… | top users | synonyms

3
votes
1answer
192 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.
3
votes
1answer
85 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 ...
-5
votes
0answers
38 views

transition matrix [on hold]

Gene mutation. Suppose a gene in a chromosome is of type $A$ or type $B$. Assume that the probability that a gene of type $A$ will mutate of type $B$ in one generation is $10-4$ and that a gene of ...
0
votes
0answers
124 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
168 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
153 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
31 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
25 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
108 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 ...
-1
votes
0answers
47 views

The invertibility of matrix (I - XX')? [closed]

I is an identity matrix of size n*n. X is a matrix of size n*k(Assuming k<= n). As we know, (I+XX') is invertible. Because (I+XX') = (I X)*(I X)', where (I X) is invertible. So I'm wondering ...
0
votes
0answers
115 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
62 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
131 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
92 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
519 views
+100

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 ...
1
vote
1answer
61 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
138 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 ...
4
votes
1answer
123 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) = ...
0
votes
1answer
81 views

How to convert non-PSD matrix to PSD matrix?

I have a mixed-integer optimization problem with the following constraint matrix $Q_1$: \begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & -1 & 0 ...
1
vote
1answer
81 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
146 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
244 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: ...
1
vote
2answers
138 views

Characteristic polynomial of Kronecker/tensor product

This was asked before on stackexchange but no answer was given. The question is the following: Let $A$ and $B$ be matrices in $GL(n)$ and $GL(m)$ respectively. Their tensor product $A\otimes B$ is ...
0
votes
0answers
26 views

Estimate bounds on Minkowski distance from point to a segment in Lp space

Assumptions Let $L_p(x,y)=(\sum_i|x_i - y_i|^p)^{1/p}$ (Minkowski metric), $a,b$ be arbitrary $n$-dimensional points, $c$ be a point that satisfies $L_p(a,b) = L_p(a,c) + L_p(c,b)$, i.e., a point ...
1
vote
1answer
60 views

When a homogeneous map between vector spaces is also additive?

Suppose to have two real vector spaces $V$ and $W$ and an injective map $T:V\rightarrow W$ such that $T(\alpha v)=\alpha T(v)$ for all $v\in V$ and $\alpha \in\mathbb{R}$. Do there exist some ...
3
votes
0answers
60 views

Reasoning about dependent and independent quantities by “degrees of freedom”

In his classic textbook Foundations of the Theory of Probability Kolmogorov defines Independence a little bit differenent then it is usually done today. He denotes a probability space by $(E, \mathcal ...
2
votes
2answers
120 views

Number of *distinct* dot products of an integer vector by elements of a hyper-rectangle

Imagine a vector $\boldsymbol{v}$ composed of integers, and the set $S$ of all integer vectors within a hyper-rectange, with one corner at the origin and other at $\boldsymbol{m}$. In other words: $S ...
4
votes
2answers
111 views

Graphs whose degree vectors coincide for all powers of their adjacency matrices

Let symmetric $A,B \in \{0, 1\}^{n \times n}$ denote the adjacency matrices of two simple graphs. Further let $\mathbf{1}$ denote the all-one-vector. Now assume that $A^k \mathbf{1} = B^k \mathbf{1}$ ...
2
votes
0answers
47 views

Sum of the entries of the inverse covariance matrix

Let $T \in\left(0,1\right)$, $n\in\mathbb{N}$ and $e_n = [1,\ldots,1]\in\mathbb{R}^n$. Consider the covariance matrix $\mathfrak{A}_n = ...
-2
votes
0answers
75 views

Minimum distance in unit cube

Given the unit cube $I^n=[0,1]^n$, let $x_1,\cdots,x_n\neq (0,0,\cdots,0)$ be $n$ linearly independent vertices (corners) of $I^n$, and $p(x_1, \cdots, x_n)$ be the projection of $(0,0,\cdots,0)$ onto ...
2
votes
1answer
95 views

Quaternion Wishart matrices of half-integer dimension?

For a physics application (quantum delay times of a chaotic scatterer) I need to generate $m$ positive random variables $\lambda_1,\lambda_2,\ldots\lambda_m$ with probability distribution ...
0
votes
0answers
54 views

Computer software for manipulating loop groups or matrices with polynomial entries

I need to deal with loop groups $LG$ over the complex numbers $\mathbb{C}$, as well as related spaces like the affine Grassmannian and affine flag variety a lot. In type A, the loop group consists ...
5
votes
0answers
115 views

Perron-Frobenius theory for reducible matrices

Can someone suggest some sources/references dealing with the Perron-Frobenius theory for nonnegative matrices that are reducible? Specifically, if $A\ge 0$ is a $d\times d$ matrix with no assumptions ...
0
votes
0answers
42 views

Computer Program for Calculations in Tensor Algebra Quotients [migrated]

Let $V$ be a finite dimensional vector space, and $S$ a finite dimensional subspace of its tensor algebra ${\cal T}(V)$ for which $X := {\cal T}(V)/<S>$ is finite dimensional, where $S$ is the ...
4
votes
1answer
152 views

Resolvent of a triangular matrix

Suppose $A$ is a triangular matrix. What is the most efficient known algorithm to compute the polynomial (in $x$) matrix $(xI-A)^{-1}$? Of course, $(xI-A)^{-1}= N(x)/p_A(x)$, where $p_A$ is the ...
3
votes
1answer
189 views

Example of proof using the generic matrix

There is a really nice proof of the Cayley-Hamilton Theorem using the generic matrix. I expose it briefly. One defines the generic matrix $G:=(X_{ij})_{ij} ...
3
votes
0answers
90 views

Worst-Case Solution to (Stochastic) Matrix Inequality

EDIT: Some specific conjectures added. This problem comes with an associated stochastic process, but I phrase everything as linear algebra in case somebody from a non-probability community has seen ...
-1
votes
1answer
118 views

Powers of orthogonal matrices is closed [closed]

This might be a basic question, nonetheless I cannot give a proof. Given an orthogonal matrix $A$ with eigendecomposition $A = Q \Lambda Q^{-1}$ with only non-real eigenvalues. Given also a diagonal ...
0
votes
0answers
69 views

Is there an improvement for the Schur-Horn inequalities for positive semi-definite matrices?

By the Schur-Horn inequality I am thinking of the statement that for any Hermitian matrix $H$ its diagonal n-tuple $(H_{11},H_{22},..,H_{nn})$ for any choice of basis lies in the convex hull of the ...
0
votes
0answers
180 views

Eigenvalue of a linear map over finite field

Let $ F_q $ be a finite field with $ q $ elements. Let $ g $ be a multiplicative generator of $ F_{q^2}^* $. It implies that $ <g^{q+1}> = F_q^* $. Let $ l $ be a prime greater than $ q^2-1 ...
2
votes
1answer
108 views

Unitary factor in polar decompositions

Let $A, B$ be $n$-square (Hermitian) positive definite matrices. Let $AB=U|AB|$ be the polar decomposition of $AB$. So $U$ is unitary (called the unitary factor of $AB$). What is the optimal constant ...
9
votes
1answer
164 views

Powers of traces, integrals over spheres and class functions

I asked this on math.StackExchange a while back but got no answers. I hope I'll be forgiven for the double post. Let $V$ be a complex vector space of dimension $\operatorname{dim}_{\mathbb C} V = n$, ...
0
votes
0answers
60 views

Matrix equation

Let $A$ be $k\times n$ matrix i.e., $A=(a_{1},\ldots, a_{n})$ where $a_{j} \in \mathbb{R}^{k}$, $rank(A)=k$ and $1\leq k \leq n$. Let $q=(q_{1},\ldots, q_{n})\in\mathbb{R}^{n}$ be such that ...
2
votes
1answer
78 views

Do the sequences with divergent associated $\zeta$-function form a vector space?

Let $V$ be the set of sequences $a \in\mathbb{R}^\mathbb{N}$ such that $\lim_{n\to\infty} a_n = 0$. The set $V$ can be seen as a real vector space, with pointwise addition and scalar multiplication. ...
2
votes
0answers
123 views

Can anyone help me deduce a matrix inequality?

The following lemma is taken from references firstly. Lemma 1 [1-2] Given matrices $Q=Q^{T} , F, M$ and $N$ of appropriate dimensions, then $$Q+MFN+N^{T}F^{T}M^{T}<0$$ for all $F$ satisfying ...
6
votes
0answers
71 views

In what sense is the Bayesian posterior mean a “convex combination”?

I asked this on math.stackexchange with no response, I'm hoping someone here might have something. Suppose I want to estimate $x \in \mathbb{R}^n$ from two signals with zero mean, normally ...
1
vote
0answers
91 views

Infinite matrices with a finite number of non-zero values on each row

The little bit of literature on infinite matrices I've been able to find studies a general setting in which the theory is hindered by constantly having to worry about whether or not various infinite ...
1
vote
2answers
283 views

How to prove that a kernel is positive definite?

For example, how to prove $\forall(x,y)\in R^N\times R^N,K(x,y) = \displaystyle\frac{1}{1+\frac{||x - y||^2}{{\sigma}^2}}\\$ where $\sigma > 0$ is a parameter, is positive definite? I have tried to ...
-3
votes
3answers
292 views

Determinant of matrix from set {-1, 1} [closed]

Let $A \in \mathbb{R}^{11 \times 11}$ and it's elements are form set $\{ -1,1 \}$. $\mathbb{P}(-1) = \mathbb{P}(1) = 0.5$. What is a probability to get such a matrix, that $\det A > 4000$? I have ...
3
votes
1answer
101 views

Characterizing space that preserves positive-definiteness property

Given a symmetric positive-definite matrix $\Sigma$, consider the space $\mathcal{D}$ of diagonal matrices such that $\forall D\in\mathcal{D}$, the matrix $\Sigma-D\Sigma^{-1}D$ is positive definite. ...