All Questions
Tagged with fa.functional-analysis linear-algebra
325 questions
2
votes
1
answer
547
views
Shift-invariant spaces
We can define a shift-invariant space as
$$V_{\varphi}(\mathbb{Z}):=\left\{\sum_{k\in\mathbb{Z}}c_k\varphi({\cdot}-k):(c_k)\in \ell_2\right\},$$
where convergence of the series is taken to be in $L^2(\...
- 49
2
votes
1
answer
145
views
Expected matrix created from two random orthogonal-projection matrices
Consider an arbitrary finite set of orthogonal-projection matrices (symmetric, idempotent, etc.) in $\mathbb{R}^{n\times n}$.
We draw two matrices $Q,P$ uniformly and i.i.d. from this set.
Question: ...
- 673
2
votes
0
answers
267
views
Example of a unital contractive map that is not completely positive on an operator system
I am aware of maps that are positive but not completely positive (for example transpose map). BUT I can not think of an example of the following type.
Does there exist an operator $T$ such that a map $...
- 231
2
votes
0
answers
81
views
Perturbed Gram matrix
Let $x_t \in \mathbb{S}^{d-1}$, $\forall t\in \mathbb{N}$ and let $e_1$ be the first canonical basis vector of $\mathbb{R}^d$, ie, $e_1 = (1,0,\cdots,0)$. Let us form a Gram Matrix
$$\sum_{t=1}^T(x_t ...
- 215
2
votes
0
answers
66
views
Minimizing a certain norm of the identity operator on $\mathbb R^2$
$\newcommand\R{\mathbb R}\newcommand\Q{\mathcal Q}$For mutually orthogonal vectors unit vectors $a=[a_1,\dots,a_n]^T$ and $b=[b_1,\dots,b_n]^T$ in $\R^n=\R^{n\times1}$ (so that $n\ge2$) and for all $x=...
- 128k
2
votes
0
answers
99
views
Lower bound on iterated matrix application
Let $n \in \mathbb Z^2$ such that the non self-adjoint weighted Laplacian is
$$(\Delta u)(n)=u(n_1+1,n_2)-u(n_1-1,n_2) + i( u(n_1,n_2+1)- u(n_1,n_2-1))$$
the adjoint operator is then
$$(\Delta^* u)(n)=...
- 192
2
votes
0
answers
114
views
Is $A$ is small on bounded functions, is there a large subdomain on which $A$ is small?
Let $A$ be a symmetric linear operator of norm $\leq 1$ on the space of functions $f:S\to \mathbb{R}$, where $S$ is a set with $N$ elements. Define the inner product $\langle \cdot,\cdot\rangle$ by ...
- 20.2k
2
votes
0
answers
58
views
Is keeping the kernel fixed an open condition for maps of vector bundles?
More precisely, let $M$ be a smooth manifold, $E_1$, $E_2$ vector bundles over $M$, and consider a $C^\infty(M)$-linear map $A:\Gamma(E_1) \to \Gamma(E_2)$ of vector bundles.
Now consider the ...
- 73
2
votes
0
answers
136
views
Linear independence of functions
Let $x_1,x_2,\ldots,x_n\in\mathbb{R}^d$ be points so that no one point is in the positive span of another. That is, there is no pair of points $x_i,x_j$ such that $x_i=\alpha x_j$ for a positive ...
- 859
2
votes
0
answers
240
views
Discrete Sobolev embedding
It is true in one dimension that $H^1$ is continuously embedded in $L^{\infty}.$
Now, consider a compact interval $[0,1]$ with a partition $I_n:=([m/n,(m+1)/n])_{m \in \left\{0,...,n-1 \right\}}$ and ...
2
votes
0
answers
57
views
The significant role of dual frames in the progress of Frame theory
For a given frame $\{\zeta_i\}_{i=1}^\infty$, any Bessel sequence $\{\eta_i\}_{i=1}^\infty$ satisfying in the following identity for every $\xi\in H$
$$\xi=\sum_{i=1}^\infty \langle \xi, \eta_i\...
- 21
2
votes
0
answers
256
views
The nonlinear operator defined as the commutator of a matrix and a nonlinear operator
In my studies of applied analysis and applied linear algebra, this interesting problem and concept came up:
Let us consider the space of all $ m \times n $ real matrices, and define a scalar ...
- 620
2
votes
0
answers
246
views
Decay rate of least eigenvalue of Gram matrices
Consider the Hilbert space $H=L^2_w(I)$ as the weighted $L^2$ space, where $I\subseteq\mathbb{R}$:
$$ L_w^2(I)=\{\phi:I\rightarrow\mathbb{R}:\,\|\phi\|^2=\int_I \phi(x)^2w(x)\,dx<\infty\}. $$
In ...
- 141
2
votes
0
answers
92
views
Estimating the size of a subset of $\mathbb{R}^N$
This concrete geometric question has arisen out of the problem of counting arithmetic functions with a particular property. The details of the relationship between the counting procedure and this ...
- 2,480
2
votes
0
answers
147
views
Is the following inequality true for the norm of Moore-Penrose pseudoinverses?
Let $L$ be a real, positive semi-definite, symmetric, square matrix, with pseudoinverse $L^{+}$. It can be shown for the operator norms $||.||_{op}$ that: if $L$ is invertible and $||I - L||_{op} < ...
- 1,465
2
votes
0
answers
463
views
Conditions for continuity of non-simple eigenvectors
Here, https://math.stackexchange.com/a/1146455, it is noted that eigenprojections are continuous, but eigenvectors are not. Are there any conditions where the eigenvalues are not simple, but the ...
- 37
2
votes
0
answers
125
views
When is an selfadjoint operatorvalued matrix with positive semidefinite diagonal elements positive semidefinite as well?
We have $p \in \mathbb{N}$ and $\mathcal{H}$ is a Hilbert space.
let's consider a matrix $\boldsymbol{\Gamma}_p := (C_{i-j})_{i,j=1, ..., p} \in \mathcal{S_H}^{p\times p}\!\!\,,$ that is a $p\times p$...
- 195
2
votes
0
answers
216
views
Separating duality for TVS?
What is the modern concept (term) for "separating duality" (dualité séparante in french) in the sense of Bourbaki (TVS Ch II § 6) as explained in the following ?
...
- 3,738
2
votes
0
answers
88
views
System of 2 linear q-difference equations with singular matrix
I would like to solve the following algebraic linear system of q-difference functional equations:
\begin{cases}
a_{11}\left(x\right)f\left(x\right)+a_{12}\left(x\right)g\left(x\right)=f\left(qx\right)...
- 133
2
votes
0
answers
648
views
Closed-form expressions for dual norms of real normed vector spaces
Didn't get any biters over at MSE, so I figure this place might be more appropriate...
Say that $V$ is a finite-dimensional real normed vector space, where for some $v \in V$ the norm is notated by $\...
- 4,897
2
votes
0
answers
520
views
Eigenvector of infinite matrix
I consider the system of reaction-diffusion PDEs in a ball
with Robin boundary condition.
It is a Steklov eigenvalue problem
(see G Auchmuty (2004) "Steklov eigenproblems and the representation
of ...
- 31
1
vote
2
answers
251
views
Linear independence of translates of a function $\{ \phi(\cdot - x) : x \in \mathcal F \}$
Let $0 \neq \phi \in L^2(\mathbb R^n)$ be a square-integrable function and $\mathcal F \subset \mathbb R^n$ a finite set. If we are in the one-dimensional setting $n=1$ then the set of translates of $\...
- 437
1
vote
1
answer
119
views
Eigenvalues under linear transformation
Let $X$ and $Y$ be square non-symmetric matrices of the same size. Assume that their eigenvalues are close in the sense that there exists a small $\varepsilon>0$ such that, for any eigenvalue $\...
- 31
1
vote
1
answer
220
views
Dimension of commutant
Suppose that $A = M_n(\mathbb{C})$ be the algebra of $n*n$ matrices over $\mathbb{C}$.
If com(A) = {$B \in M_n(\mathbb{C}); AB = BA$}, then what is the $dim(com(A))?$
- 581
1
vote
1
answer
1k
views
Inequality for the operator norm of a product of matrices
I am working with a product of $n\times n$ matrices $A_1,\ldots,A_k$. Under which conditions can I assume that
$$\|A_1\cdots A_k\|_\infty \leq \|A_1\cdots \hat{A_i}\cdots A_k\|_\infty \|A_i\|_\infty,...
- 449
1
vote
1
answer
387
views
All and the only algebraically closed fields s.t. any regular n-by-n matrix has a k-th root for every k
The title has it all. I'm looking for a proof/disproof of the fact that an algebraically closed field, say $\mathbb K$, has characteristic zero iff the following property (R) holds: For all $n,k \in \...
- 10.5k
1
vote
1
answer
304
views
How do maximum norms relatively change in Euclidean translations
Let $Q$ be the cube $[-1,1]^{3}$ and $\pi$ be a plane in $\mathbb{R}^{3}$
that contains the origin but doesn't contain any vertex of $Q$. Suppose that $A$ is an invertible
linear transformation from $\...
- 11
1
vote
1
answer
158
views
When do we have $B_Y\subset T(B_X)$ if and only if $\overline{B_Y}\subset T(\overline{B_X})$?
Let $X$,$Y$ be normed spaces, $T:X\to Y$ be a bounded linear operator. Denote the open and closed unit balls by
$$
B_X:=\{ x\in X\ |\ \|x\|<1\} \\
\overline{B_X}:=\{ x\in X\ |\ \|x\|\le1\}
$$
and ...
- 1,245
1
vote
3
answers
684
views
Norm of an operator formed using a unitary operator
Suppose, $ A $ is a unitary matrix in $ M_n(\mathbb{C}) $ given by $ (a_{i,j})_{1\le i,j\le n} $ which has the property that, for all the basis elements $ e_i $, $ Ae_i\ne |\lambda| e_j $ for all $i,j ...
- 817
1
vote
2
answers
289
views
Any analysis on phase of eigenvalue of unitary matrix?
I understand that there are invariant Haar measure for eigenvalues of unitary matrix. I further understand that absolute value of eigenvalues of unitary matrix is 1. But, I could not find any analysis ...
- 11
1
vote
1
answer
155
views
Spectrum invariant under (generalised) transpose as operator on trace class operators
For matrices $A$ it is well known that the spectrum is invariant under transpose $\sigma(A^T) = \sigma(A)$. Furthermore, the spectrum of the adjoint matrix $\sigma(A^*) = \overline{ \sigma(A)}$ the ...
- 1,054
1
vote
1
answer
322
views
A particular commutator of the discrete Fourier matrix
For $N$ be a fixed natural number, define $w=e^{\frac{2\pi i}{N}}$ and $z=e^{\frac{\pi i}{N}}$, so that $z^2=w$. Let $D$ be the diagonal matrix $D=\operatorname{diag}(1,z,z^2,\ldots,z^{N-1})$ and $F$ ...
- 4,058
1
vote
1
answer
195
views
Eigenvalues of operator
In the question here
the author asks for the eigenvalues of an operator
$$A = \begin{pmatrix} x & -\partial_x \\ \partial_x & -x \end{pmatrix}.$$
Here I would like to ask if one can extend ...
- 192
1
vote
2
answers
693
views
Separating convex sets in Vector spaces
This question just popped on my mind.
Let $A, B$ two disjoint, nonempty convex sets in the vector space $X$, can they be separated via a nonzero linear function in $X' = \{ f : X \to R ~ | \quad \...
- 369
1
vote
1
answer
685
views
dual space of the quotient space of some locally convex topological space
I would like to a classical result about dual space. Let $E$ be a locally convex space and $F$ its closed linear subspace. If $E^{\ast}$ is the dual space of $E$, could some one affirm me that the ...
- 1,835
1
vote
1
answer
223
views
A linear algebraic q-difference equation [SOLVED]
I would like to solve the following algebraic linear q-difference equation:
\begin{equation}
a\left(x\right)f\left(x\right)=f\left(qx\right)
\end{equation}
The parameter $q$ is real, positive and ...
- 133
1
vote
1
answer
254
views
references for families of conditionaly negative definite matrices
We say that a matrix $A\in M_n(\mathbb{C})$ is a conditionaly negative definite matrix if it is hermitian and if for all complex numbers $c_1,\ldots,c_n$ such that $c_1+\cdots +c_n=0$ we have
$$
\sum_{...
- 1,222
1
vote
1
answer
91
views
Positive definite kernels on compact interval $[0,1]$
From How to prove that a kernel is positive definite? I learned that a function $f:[0,\infty)\to\mathbb{R}$ induces a positive definite kernel $K:\mathbb{R}^2\to\mathbb{R}$, $K(x,y)=f((x-y)^2)$ if $f$ ...
- 216
1
vote
1
answer
142
views
Operator norm of some type of discrete Fourier matrix
Let $N$ be a natural number and let $w$ be a complex number.
We define the $N\times N$ matrix $C_w=(a_{k,l})_{k,l=1}^N$ as follows,
$$
a_{k,l}=\begin{cases}1 & l=k+1\\
w &...
- 4,058
1
vote
1
answer
179
views
Definition and properties of tangent functional
I am reading Measures Which Agree on Balls by Hoffmann-Jørgensen and I am somewhat confused. Here, $E$ is a Banach space, $S$ is the unit sphere, and $x \in S$.
We let $\tau(x, \cdot)$ denote the ...
- 297
1
vote
1
answer
114
views
Is a $1_A \otimes U$ invariant subspace of $\mathcal{H}_A \otimes \mathcal{H}_B$ a product $V_A \otimes \mathcal{H}_B$?
Consider a subspace $V$ of $\mathcal{H}_A \otimes \mathcal{H}_B$, with $\mathcal{H}_A$ and $\mathcal{H}_B$ finite-dimensional Hilbert spaces, that is $1_A \otimes U$ invariant for all unitary ...
- 133
1
vote
1
answer
100
views
$\ell^1$-bound on graph laplacian with weight
Consider the $\mathbb Z^2$ lattice, we then define for $u=(u_{ij})_{i,j \in \mathbb Z}$ the discrete Laplacian
$$(\Delta u)_{i,j}=u_{i+1,j}+u_{i-1,j}+ u_{i,j+1}+u_{i,j-1}$$
and the weight which pushes ...
- 536
1
vote
1
answer
136
views
Conditions to obtain a real logarithm of a unitary unimodular complex matrix?
The problem statement is the following:
$$U=\exp\{iV\}$$
where $U$ is a unitary unimodular matrix of the following form:
$$U=\begin{bmatrix}u_1+iu_2&u_3+iu_4\\-u_3+iu_4&u_1-iu_2\end{bmatrix}...
- 113
1
vote
2
answers
400
views
Reference request for Stieltjes Transform
I am wondering to use Stieltjes transform to signal processing like Fourier Transform. The Fourier is known to give the frequencies of a signal but not sure what Stieltjes transform gives. I am only ...
- 27
1
vote
1
answer
1k
views
The norm of a Finite Hilbert matrix
Let $H$ be an $n\times n$ Hilbert matrix,
$$h_{ij}=(i+j-1)^{-1}.$$
The matrix $p$-norm corresponding to the p-norm for vectors is:
$\left \| A \right \| _p = \sup \limits _{x \ne 0} \frac{\left \|...
- 1,748
1
vote
1
answer
176
views
Linear map with two "incompatible" representations
Let $K$ be a field and let $V$ be the set of sequences $\{v_1,v_2,\dots\}$ of elements of $K$. If $A=\{a_1,a_2,\dots\}$ is also a sequence of elements of $K$, then it defines an endomorphism of $V$ $$...
- 6,924
1
vote
1
answer
151
views
Is smoothness preserved under an isometric isomorphism?
Let $(X, \|.\|_1)$ is isometrically isomorphic to $(X, \|.\|_2)$ and $\|.\|_2\leq \|.\|_1$. Assume that $x_0$ is a smooth point of $(X, \|.\|_1)$ and $\|x_0\|_2=1$. According to the definition of a ...
- 113
1
vote
1
answer
291
views
Norm of solution of quadratic program
In a quadratic program (QP), do linear equality constraints always reduce the norm of the minimizer? Specifically, let $P \succ 0$, $A \in \mathsf{M}_{m\times n}$ and $q\in\mathbb{R}^n$. Define
$$x^* ...
- 153
1
vote
1
answer
136
views
Self adjoint operator and vertex conditions in quantum graphs
Let $\Gamma$ be a metric graph with finitely many edges. Consider the operator H acting as $\frac{-d^2}{dx_e^2}$ on each edge $e$, with the domain consisting of functions that belong to $H^2(e)$ on ...
- 11
1
vote
1
answer
225
views
About expectation norms on graphs
Let $S \subseteq V$ of a $d-$regular graph $G$ such that $\mu = \frac{\vert S \vert }{\vert V \vert } $. Let $A$ be the adjacency matrix of the graph. Then define the quantity $\phi(S)= \frac{E(S,\bar{...
- 1,893