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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)=...
Kung Yao's user avatar
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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 ...
H A Helfgott's user avatar
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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 ...
user337331's user avatar
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 ...
mohi's user avatar
  • 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 ...
AlgebraicGeometer's user avatar
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\...
Javani's user avatar
  • 21
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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 ...
groupoid's user avatar
  • 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 ...
user39756's user avatar
  • 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 ...
Kevin Smith's user avatar
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2 votes
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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} < ...
Learning math's user avatar
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 ...
billbob's user avatar
  • 37
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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 $p\times p$...
Obriareos's user avatar
  • 195
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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 ? ...
Duchamp Gérard H. E.'s user avatar
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)...
user2983638's user avatar
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 $\...
Mike Battaglia's user avatar
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 ...
Szopa's user avatar
  • 31
1 vote
0 answers
44 views

Characterizing some similarity invariant homogeneous log-superharmonic functions of matrices

Let $L:M_n(\mathbb{C})^r\rightarrow[0,\infty)$ be a function that satisfies the following properties: $\log(L)$ is plurisubharmonic. $L$ is homogeneous in the sense that $L(\lambda A_1,\dots,\lambda ...
Joseph Van Name's user avatar
1 vote
0 answers
133 views

Infinite dimensional matrix solvability

In order to solve a boundary problem for a conducting hemisphere, the following matrix equation arises (derived from the boundary condition on the curved part of the surface) where we must solve for ...
Matt Majic's user avatar
1 vote
0 answers
87 views

Continuous choice of null directions for a family of bilinear forms

Let $E$ and $F$ be (real) Hilbert spaces, where $\dim E = \infty$ and $1 \leq \dim F < \infty$. Let $T : E \to \operatorname{Sym}(F \times F,\mathbb{R})$ be a continuous linear map, where $\...
Eduardo Longa's user avatar
1 vote
0 answers
64 views

Small perturbation to a commuting family of hermitian matrices will hurt the nice properties?

Let $A_1, \dotsc A_N$ be a collection of finite Hermitian matrices that commute with one another and all have the matrix $2$-norm as $1$. Here $N$ is large but fixed. Then, they are simultaneously ...
Isaac's user avatar
  • 3,477
1 vote
0 answers
87 views

Computation of the trace of a convolution operator

I try to understand a particular step on page two from the research paper "A multidimensional analog of a limit theorem of G. Szegö". https://iopscience.iop.org/article/10.1070/...
HyyFly's user avatar
  • 197
1 vote
0 answers
61 views

Discrete Orlicz space estimate

We consider the discrete LlogL space of sequences $x=(x_i)$ such that $$\Vert x\Vert_{LL}:=\sum_i \vert x_i \log(x_i)\vert <\infty.$$ Let $x=(x_i)$ and $y=(y_i)$ two sequences in the above LlogL ...
The entropy method's user avatar
1 vote
0 answers
70 views

Lower-bounding the eigenvalues of a certain positive-semidefinite kernel matrix, as a function of the norm of the input matrix

Let $\phi:[-1,1] \to \mathbb R$ be a function such that $\phi$ is $\mathcal C^\infty$ on $(-1,1)$. $\phi$ is continuous at $\pm 1$. For concreteness, and if it helps, In my specific problem I have $\...
dohmatob's user avatar
  • 6,853
1 vote
0 answers
448 views

Smallest eigenvalue for large kernel matrix

I am interested in the the asymptotics of the minimum eigenvalue $\lambda_n^n$ of a class of kernel matrix $P = [ K(x_i - x_j) ]_{i,j}$, with $x_i$ equally spaced in the unit cube of $\mathbb{R}^d$. ...
KDD's user avatar
  • 151
1 vote
0 answers
48 views

Is there an easy characterisation (perhaps some generalised Löwner representation) for operator monotone functions of order $n$?

As per my understanding, roughly stated, $f$ is an operator monotone function of order $n$ if for all $n\times n$ (Hermitian) matrices, $X,Y\ge0$ which satisfy $X\ge Y$, we have $f(X)\ge f(Y)$. If $f$...
Atul Singh Arora's user avatar
1 vote
0 answers
54 views

When does the multi-spectral radius coincide with the spectral radius of the sum of linear transformations?

Suppose that $X$ is a finite dimensional Hilbert space and $A_{1},\dots,A_{r}:X\rightarrow X$ are linear transformations. Define the multi-spectral radius $\rho(A_{1},\dots,A_{r})$ to be $$\limsup_{n\...
Joseph Van Name's user avatar
1 vote
0 answers
127 views

What is the analogue of expansive matrix for automorphisms?

We say an invertible $n \times n$ matrix with entries in $\Bbb R^n$ is expansive if the absolute values of all of its eigenvalues exceed $1$. An easy calculation also shows that if we consider a ball ...
Melody's user avatar
  • 41
1 vote
0 answers
94 views

Space spanned by pointwise squares of basis functions

Consider the Hilbert space $L^2(\Omega)$ over some Euclidean domain $\Omega$. Let $F=\{f_i;i\in\mathbb N\}$ be an orthonormal basis of this space consisting of functions in $L^2(\Omega)\cap L^4(\Omega)...
Joonas Ilmavirta's user avatar
1 vote
0 answers
256 views

Significance of Tikhonov matrix

I am looking for a tutorial on Tikhonov matrix, in the sense what it can do or it cannot do. The definition of the matrix can be obtained in the wikipedia link. https://en.wikipedia.org/wiki/...
lordOfTheRings's user avatar
1 vote
0 answers
83 views

What is optimal distance between inverse of convolution operator?

I am looking for a measure to find the optimal distance measure between inverse of an convolution operator $A$ and say another convolution operator $B$. I want my measure to be sharp that mean when $B$...
lordOfTheRings's user avatar
1 vote
0 answers
174 views

Negative eigenvalue of Toeplitz Hermitian matrix?

I am working on estimation of a covariance matrix and I know that the matrix is Toeplitz. The desired matrix should not produce negative eigenvalues at all. However, sometime my estimation leads to a ...
Creator's user avatar
  • 495
1 vote
0 answers
85 views

What are good bounds on ratios of subdeterminants?

Let $A$ be a symmetric matrix and $A_i$ be the matrix obtained from $A$ by dropping the $i^{th}$ row and column. Then what are some good bounds on the value of $\frac{det(A_i)}{det(A)}$ ? Using the ...
user6818's user avatar
  • 1,893
1 vote
0 answers
270 views

Eigenvalue of product of self adjoint compact operators

Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...
Madhuresh's user avatar
  • 157
1 vote
0 answers
147 views

Bounding Rayleigh quotient for stochastic matrix

Suppose you have an irreducible, stochastic matrix $A$ with left Perron-Frobenius eigenvector $v$ (corresponding to the eigenvalue $1$), and suppose the next largest eigenvalue for $A$ is $\lambda$. ...
Rookatu's user avatar
  • 121
1 vote
0 answers
174 views

Eigenvalues of a Parametrized Family of Linear Functions

Suppose that we have a family of linear functions $L(\alpha) : \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $\alpha$ is a positive real number. For each $\alpha$, it is given that $L(\alpha)$ is a ...
Eric Haengel's user avatar
1 vote
0 answers
466 views

Bounding point-wise maximum of the absolute difference of two convex functions

Let $\Delta: R \times R \rightarrow R_{+}$ be a positive and convex function (convex in, say, both the arguments) called the loss function. Let $x \in R^d$. Moreover, let $H_1,...,H_r$ be sets of ...
Rajhans's user avatar
  • 11
0 votes
0 answers
66 views

Taking trace of a tensor product of matrix-valued smooth functions on the thin diagonal

Let $V$ be a finite dimensional real / complex vector space and consider the space $L(V,V)$ of linear operators on $V$. Fix $n \in \mathbb{N}$ and let $\mathcal{M}$ be the real / complex vector space ...
Isaac's user avatar
  • 3,477
0 votes
0 answers
46 views

What's the problem in using spanning Bessel sequences that are not frames to decompose vectors?

This is related to a question I recently asked on math.SE. Consider a subset $G\equiv \{g_k\}_{k\in\mathbb{N} }\subseteq\mathcal H$ in a separable Hilbert space $\mathcal H$, and suppose $G$ spans the ...
glS's user avatar
  • 342
0 votes
0 answers
93 views

Orthogonalization of symmetric non-degenerate bilinear forms

It is well-known that given a field $k$ with characteristic different from $2$, every symmetric non-degenerate bilinear form $B$ over a finite-dimensional space can be orthogonalized. This means that ...
Luiz Felipe Garcia's user avatar
0 votes
0 answers
55 views

Johnson-Lindenstrauss type result for matrix factorization

The type of result I want is: given matrix $A\in \mathbb{R}^{m\times n}$ and error tolerance $\epsilon$, what is a lower bound on $k$ such that $\|A - UV\|_{??}\le \epsilon$, where $U \in\mathbb{R}^{m\...
optimal_transport_fan's user avatar
0 votes
0 answers
64 views

When is a symmetric block Toeplitz matrix invertible?

Let $$ Q = \begin{bmatrix} Q_0 & Q_1 & Q_2 & \cdots\\ Q_{-1} & Q_{0} & Q_1 & \cdots\\ Q_{-2} & Q_{-1} & Q_0 & \cdots\\ \vdots & \vdots & \vdots & \ddots ...
Benjamin Tennyson's user avatar
0 votes
0 answers
163 views

Generalization of polynomial coefficients

I'm dealing with a hard combinatorial problem where for every positive integer value of a variable $n$ I have to calculate a list of numbers, specifically $n^2$, that depend on $n$ and its list index ...
Cardstdani's user avatar
0 votes
0 answers
74 views

Computing the eigenvalues of $A+E$ where $A$ is an upper triangular matrix whose diagonal entries are all zero and $E$ is a rank one matrix

Let us consider the backward-shift matrix $B=(b_{ij})\in M_n(\mathbb{R})$ whose entries are given by $b_{k,k+1}=1$ and the other entries are all 0. We also consider $X=(x_{ij})\in M_n(\mathbb{R})$ ...
ABB's user avatar
  • 4,058
0 votes
0 answers
146 views

Linear dynamics in a function space

I posted the same question to Math Stackexchange earlier without much luck, so I am posting here. I am dealing with a time-dependent model, which can be expressed as a function. $f$ is dependent on ...
CWC's user avatar
  • 433
0 votes
0 answers
124 views

Eigenvectors of the symmetric tridiagonal matrices whose entries above the diagonal are all the same

Let us consider the real symmetric tridiagonal matrix $T=(t_{kl})$ in $M_n(\mathbb{R})$ with $$t_{1,2}=t_{2,3}=\cdots=t_{n-1,n}=1$$ How can we compute the eigenvectors of $T$?
ABB's user avatar
  • 4,058
0 votes
0 answers
272 views

Finding the eigenvectors of a submatrix

Let $A=(a_{kl})$ be a matrix in $M_n(\mathbb{R})$ when $n$ is even. Let $B=(b_{kl})$ be the symmetric $2n$ by $2n$ matrix whose entries are given by, $b_{k,l}=a_{kl}$ if $1\leq k,l\leq n$. $b_{n+k,l}=...
ABB's user avatar
  • 4,058
0 votes
0 answers
104 views

Necessary and sufficient conditions for $\mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}) \ge 0$ for all $t$

Let $A$ and $B$ be positive-definite matrices of the same size. For any $t \ge 0$, define $$ u(t) := \mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}). $$ Question. What are necessary and ...
dohmatob's user avatar
  • 6,853
0 votes
0 answers
174 views

Lipschitz map on positive definite cone of $n$-by-$n$ matrices

A function matrix $f : X \to \mathbb R$ is a convex Lipschitz continuous matrix function with Lipschitz constant $\mathrm L$ with respect to a fixed given norm $\|\cdot\|$, i.e., $|f(A)-f(B)| \leq \...
Reza's user avatar
  • 91
0 votes
0 answers
92 views

Positive definite matrix and Hörmander theory

Let $\varphi \in C_{0}^{\infty}, \varphi\neq 0$. We'll consider the inner product in $L^{2}.$ Let $\alpha,\beta$ multi-index, $m\in \mathbb{N}$ such that $|\alpha|,|\beta|\leq m$ and set $$ \varphi_{\...
Lucas De Souza's user avatar
0 votes
0 answers
112 views

How to prove or disprove $ |r'(a)| \leq c_1 \sup_{x \in [a-1/2,a+1/2]} |r(x)|$?

Let $ A = \begin{bmatrix} a & 1 \\ 0 & a \end{bmatrix}$ be a Jordan matrix with $ -1 < a < 1 $. Let $r(z) = \frac{p(z)}{q(z)}$ be an irreducible rational function, where $p(...
xiuhua's user avatar
  • 101