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

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1
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0answers
35 views

Products of random permutations with fixed matrix

This question originates from an engineering problem, which I am solving. Any related references are highly appreciated. Let $M_k(T)=\prod_{t=1}^T P_t S_k$ over some field (finite or reals), where ...
2
votes
1answer
89 views

Determinant of a Certain Positive-Definite Block Matrix

Is there a lower bound for the determinant or minimum eigenvalue of the following $d$ by $d$ matrix in terms of $d$? $$\Gamma=\left( {\begin{array}{cc} I & B \\ B^{*} & I \\ \end{array} ...
2
votes
0answers
107 views

Determinant Evaluation

Is there a closed form (something involving a ratio of products) for: $$\det\left[\binom{a_i+c}{a_i-i+j}\right]_{1\leq i,j\leq t},$$ where $a_i,c$ are positive integers? I think with $c=0$ this is ...
2
votes
1answer
190 views

Classes of finite resoluble groups which are (faithfully) representable by triangular matrices?

Let $G$ be a group, $k$ a field and $T(n,k)\subset Gl(n,k)$, the group of invertible upper triangular $n\times n$ matrices. I know that if $\rho : G\rightarrow T(n,k)$ is faithful (i.e. into) then ...
6
votes
1answer
117 views

Monte-Carlo computation of the Smith normal form

Quite some time ago I saw an article where a Monte-Carlo algorithm for computing the Smith normal form of an integer matrix was described. In this article the following problem was posed: Suppose $P, ...
3
votes
1answer
253 views

simultaneous action of GL(n) on the matrices

Consider the action of GL(n,k) on the set MxM where M is the set of all n-by-n matrices over k given by $g.(h,l) \mapsto (ghg^{-1}, glg^t)$. Individually these actions are well studied and good ...
3
votes
1answer
135 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 bilinear form to be $\begin{pmatrix} O&I\\ I&0\end{pmatrix}$. Let u be a unipotent in ...
0
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0answers
39 views

Bounds on the smallest eigenvalue of a Hankel matrix

Let $H=H_n$ be a positive definite Hankel matrix of size $n$ with $\lambda_n$ is it's smallest eigenvalue. What bounds are known on $\lambda_n$ in terms of the entries on $H$. I can see some results ...
5
votes
1answer
160 views

Under what conditions a linear automorphism is an isometry of some norm?

Assume $V$ is a finite-dimensional vector space over $\mathbb{R}$, and $T: V \to V$ is a (linear) isomorphism. When is it possible to construct a norm on $V$ making $T$ an isometry? ...
27
votes
4answers
899 views

Why there is a relation among the second-order minors of a symmetric $4\times 4$ matrix?

A $4\times 4$ symmetric matrix $$ \left( \begin{array}{cccc} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{12} & a_{22} & a_{23} & a_{24} \\ a_{13} & a_{23} & a_{33} & ...
-1
votes
1answer
125 views

Orthogonal decomposition of conditional expectations

Suppose I have a random variable $x$ and a set of conditional distributions on $x$. Here is an example where the conditionals are nested: $$q_1 := E(x|y_1), \quad q_2 := E(x|y_1,y_2),\quad q_3 := ...
0
votes
1answer
48 views

Do the support sets of subspaces give the representable matroids?

Fact:   Start with $V$ a subspace of $\mathbb R^n$. Take the set of all supports of vectors in $V$. Throw out $\emptyset$. You now have the dependent sets of some matroid. Not sure you ...
0
votes
1answer
39 views

Lower bound on Spectral Gap of Rank one + Diagonal

For some $x\in\mathbb{R}^n, \|x\|_2^2=1$ and $\alpha\geq 0$, consider the positive semi-definite matrix $$ X_\alpha := xx^T + \alpha\sum_{k=1}^nx_k^2e_ke_k^T. $$ Suppose for simplicity that the ...
1
vote
1answer
88 views

Conditions for the consistency of a system of affine polynomials

Let $f_1, f_2,\ldots,f_N$ be some affine polynomials. We consider the question if these polynomials have a common (affine) root. By homogenizing these polynomials, we can associate a projective ...
0
votes
0answers
29 views

Stochastic independence of columns of projection matrix to the rest of the columns of a random matrix

First let me describe the setting of the problem. I have a random matrix $A\in \mathbb{R}^{m\times n},\ (m<n)$ with $a_{ij}\sim \mathcal{N}(0,I)$ i.i.d. Let there be a given set of $K (K<m)$ ...
0
votes
1answer
138 views

Prove that the following two optimization problems are equivalent

I am trying to solve the following optimization problem for the vector $ y $, where $ A_i $ are some given matrix (maybe low rank) and $ x_i $ are unconstrained $$ \min_{y, x_i} \sum_{i=1}^J || y - ...
1
vote
1answer
195 views

A $d$-form on ${\mathbb R}^n$ that vanishes on $\binom{d+n-1}{n-1}$ general points, vanishes identically

I'm looking for a reference for the fact that a $d$-form on ${\mathbb R}^n$ that vanishes on $p_1,..,p_{\binom{d+n-1}{n-1}}$ general points, vanishes identically. A specific construction of a set of ...
1
vote
1answer
67 views

arg min_X ||A X B - C||^2, with X diagonal [closed]

Let $A, B, C$ be known matrices, and let $X$ be an unknown matrix. Given that $C = AXB \Leftrightarrow \text{vec}(C) = K \text{vec}(X)$, where $\text{vec}(\cdot)$ denotes the vectorization of a ...
0
votes
1answer
30 views

Solving sparse linear least squares or a positive definite 5-band matrix system fast

I want to quickly solve linear least squares problem for $x \in \mathbb{R}^n$ $$min_x \left\| A x - b \right\|_2^2$$ with a special sparse structure where each row in $A$ has only up to 4 ...
9
votes
3answers
289 views

Real and Quaternionic Representations according to Weights

According to this question, it is easy to know whether a representation is self dual or not: just check if the weight distribution in space is symmetric about the origin. Now, for self dual ...
0
votes
0answers
51 views

Partial Vandermonde Circulant Determinant Expression

Consider following partial Vandermonde type, circulant matrix $\begin{bmatrix} x_1 & x_2 & 0 & \dots & 0 & x_n\\ x_1^2 & x_2^2 & x_3^2 & \dots & 0 & 0\\ \vdots ...
1
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0answers
58 views

Perturbation of eigenvalues of some special matrices

In perturbation theory of linear operators, one major question is how the eigenvalues of a linear operator $A$ change under a small perturbation, $A(x) = A + xP$, with $x\in\mathbb{R}$. For instance, ...
1
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0answers
47 views

Analogs of the paralleloram identity in higher degrees

I asked this two months ago in MSE, but nobody answered, so I hope it will be suitable here. A homogenious polynomial of degree $k\in{\Bbb N}$ on a finite dimensional vector space $X$ over $\Bbb R$ ...
7
votes
1answer
244 views

For a ring $k$ and a set $X$, what are the $k$-algebra homomorphisms $k^X \to k$?

Let $k$ be a commutative ring. Feel free to assume it's a field. Let $X$ be a set. This question is only interesting when $X$ is infinite. Write $k^X$ for the $k$-algebra of functions $X \to k$, ...
11
votes
2answers
412 views

which norms can be realized as operator norms?

Assume $(V,∥∥_V),(W,∥∥_W)$ are both finite dimensional normed spaces. We have the induced operator norm on ${\rm Hom}(V,W)$. It turns out that the operator norm is induced by an inner product iff ...
2
votes
1answer
97 views

Computation Time of Smith Normal Form in Maple

I am using maple to compute the Smith Normal Form of a matrix of size 120*120 and it seems that I will never get an answer back. I have checked my code for small cases and I believe that it is ...
1
vote
1answer
38 views

rank minimization over vector subsets

Let $S$ be a set of $n$ vectors from $\mathbb{Q}^d$. For every $k=1,2,\dots,n$, define $$r_k = \min_{T\subset S, |T|=k} \mathrm{rank}(T),$$ where $\mathrm{rank}(T)$ is the rank of a matrix formed by ...
2
votes
1answer
126 views

Is there some quantitative version of interlacing of eigenvalues of a matrix under rank-1 update?

Given a real symmetric matrix $A$ and a vector $v$ of the same dimension we know that the eigenvalues of $A + vv^T$ are left interlaced by the eigenvalues of $A$. But do we have any quantitative ...
3
votes
1answer
119 views

Condition number after preconditioning

Suppose $A$ and $P$ are symmetric, positive definite matrices and that we factor $P^{-1}=EE^\top.$ Is it true that the condition number of $PA$ is upper-bounded by the condition number of ...
0
votes
1answer
89 views

Hadamard Product and Eigendecomposition

I just found this related question in here Q1. Given a positive definite matrix $\mathbf{A}$, consider its eigendecomposition $(\mathbf{A}\mathbf{V} = \mathbf{V}\mathbf{D})$. Consider an arbitrary ...
1
vote
0answers
51 views

Epsilon-net of operator norm ball around Identity

Suppose I look at the set of matrices which are invertible and satisfy $$ \left\|A-Id\right\|_{op}<r $$ for some $r<1$, where $Id$ is the $n\times n$ identity matrix. An $\epsilon$-net of such ...
3
votes
1answer
63 views

Spectral theorem from Jordan decomposition in infinite dimensions

The finite-dimensional spectral theorem is a simple special case of the finite-dimensional Jordan decomposition. We also have various infinite-dimensional generalizations of the spectral theorem; can ...
3
votes
1answer
147 views

Eigenvectors as continuous functions of matrix - diagonal perturbations

The general question has been treated here, and the response was negative. My question is about more particular perturbations. The counterexamples given in the previous question have variations not ...
0
votes
0answers
30 views

On the induced norms of stochastic operator and its adjoint operator

The background: when studying the paper published in Automatica named '$H_{\infty}$ control and filtering of discrete-time stochastic systems with multiplicative noise' (volume 37, pp. 409-417), I ...
6
votes
2answers
203 views

What methods do we have to understand the spectrum of matrices with restricted entries?

Consider questions of the form (or the "most probable value of" version of these questions rather than the "largest possible"), What is the largest possible spectral radius of a $n \times n$ matrix ...
0
votes
0answers
46 views

Completing Karlin's proof of variation diminishing transformation theorem

In S Karlin's book total positivity there's a theorem that says if $K(x,y)$ is $TP_r$ (totally positive with degree $r$) and the sign change count of function $h$, $S(h) = n\leq r-1$, then ...
0
votes
1answer
76 views

Reference for measures of commutativity needed

I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if A and B are two n×n Hermitian matrices, and [A,B]=C. I'd like a function μ:Cn×n→[0,∞) ...
1
vote
1answer
42 views

Splines linearly independent

Let $N_1:=\chi_{[0,1]}$ be defined as this characteristic function and $N_n:=N_{n-1}*N_1$ then this leads to polynomials with support $[0,n]$. These splines are well-studied click for wikipedia My ...
6
votes
2answers
360 views

Is there a nice “synthetic” way for doing differential geometry on infinite dimensional vector spaces?

If $V$ is an infinite dimensional vector space, for example the space of smooth functions on $\mathbb{R}$, we can introduce some differential geometry concepts by choosing a topology on $V$ and doing ...
0
votes
0answers
14 views

How to treat non-identifiable states in Kalman filtering/dynamic linear models?

Let $x_t = G_tx_{t-1}+\omega_t$ with $\omega_t \sim \mathrm{N}(\mathbf{0}, \mathbf{W}_t)$ be a state equation and $y_t = F_tx_t+\nu_t$ with $\nu_t \sim \mathrm{N}(\mathbf{0}, \mathbf{V}_t)$ be a ...
4
votes
2answers
290 views

“Typical” convergence rate for the von Neumann mean ergodic theorem

The von-Neumann theorem states that for a unitary operator $U: {\cal H} \mapsto {\cal H}$, where ${\cal H}$ is a Hilbert space, the following holds: $$ \lim_{N\to \infty} \frac{1}{N} \sum_{n=1}^N U^n ...
5
votes
2answers
272 views

Minimum of squared sum minus sum of squares

I know that $$ \min_{\|x\|_2=1=\|y\|_2} \left(\sum_{k=1}^nx_ky_k\right)^2-\sum_{k=1}^nx_k^2y_k^2 \geq -1/2 $$ with equality whenever $|x_k|=\frac{1}{\sqrt{2}}=|y_k|$ for two coordinates. I'm ...
1
vote
0answers
49 views

non-intersecting families of subspaces

Given $V$, a vector space over a finite field $F$ of size $k$, if $\dim(V)=m$, and $r$ divides $m$, there exists a family of $r$-dimensional subspaces, whose size is equal to $(k^m-1)/(k^r-1)$ and ...
6
votes
1answer
210 views

Convexity of the product of two exponential matrices

Let $S\subset\mathbb{R}$ be a convex set and $\mathbb{S}^{n}$ be the set of real symmetric matrices of order $n\times n$. A matrix valued function $\Gamma: S \rightarrow \mathbb{S}^{n}$ is said to ...
1
vote
2answers
101 views

Nonlinear matrix differential equation

I want to solve the equilibrium of the following differential equation: $\dot{x_i} = \sum_j A_{ij} x_j + x_i \sum_j B_{ij}x_j$ which is essentiall in matrix notation: $\dot{\mathbf{x}} = ...
0
votes
0answers
77 views

Existing complete function space under suitable norm

This question was asked in math.stackexchange.com but no suitable answer was received, so I am posting it here. This is a question which came to me due to several previous question: sorry for the all ...
1
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0answers
92 views

What is the trace of this operator in $L^\infty$ (if this question make sense)?

Let $\Omega$ be a closed domain in $\mathbb{R}^N$, and $\lambda$ the corresponding Lebesgue measure. We define in $(L^\infty(\Omega), \Vert \cdot \Vert_\infty)$ the following operator : ...
1
vote
0answers
61 views

Smallest sum of original column entries in 2d matrix [closed]

I have an interesting optimization problem I am trying to solve now and I thought I'd share it here in order to find the best answer. The problem itself is not complicated and it is stated like this: ...
2
votes
1answer
76 views

Dimension of a certain subspace of univariate polynomials

Let $\mathbb{F}$ be an arbitrary field. For a polynomial $f\in\mathbb{F}[x]$, we use $Z(f)$ to denote set of roots of $f$ in $\mathbb{F}$. Let $S$ and $T$ be sets of elements of $\mathbb{F}$ of size ...
1
vote
1answer
61 views

Given $M$, minimize $|Mx|_0$

Given a matrix $M \in \mathbb{R}^{n \times m}$, I would like to find $\min_{x \in \mathbb{R}^m} \|Mx\|_0$ such that $x \neq 0^m$, where the $\ell_0$ "norm" is measured by simply counting the number ...