Questions tagged [perturbation-theory]

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Lipschitz continuity of eigenprojections

This question has the same flavor of this and this questions, but asks for something stronger. Assume that $A$ is a symmetric $n \times n$ matrix, $H$ is a $n \times n$ perturbation matrix. Moreover ...
Guanaco96's user avatar
1 vote
0 answers
44 views

Bounds on the spectral radius of a perturbed directed graph

Suppose $(G_n)$ is a sequence of strongly connected directed graphs (without multiple edges) with $G_n$ having $n$ edges such that the adjacency matrix $A_n$ of $G_n$ is primitive, and let $(G_n’)$ be ...
a person's user avatar
1 vote
0 answers
43 views

How to get perturbation bounds of singular vectors

Let an adjacency matrix $A={A^\top}\in {\mathbb{R}^{n \times n}}$ (a binary matrix) of a simple undirected graph and its degree matrix $D$ be given. When adding $Q$ edges into the graph, which is ...
Henry's user avatar
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2 votes
0 answers
119 views

Linear elliptic problems: Are gradient estimates preserved after perturbation?

(This question is a duplicate from here) We start with the linear elliptic PDE $$ -\operatorname{div}(A\nabla u)=f \quad\text{in}\ \Omega,\\ u=0 \quad\text{on}\ \partial\Omega $$ We assume that $\...
Muschkopp's user avatar
  • 141
1 vote
0 answers
98 views

Schrödinger equation approximation – continuity of eigenvalues with respect to potential

The question has been crossposted from Stackexchange after receiving no answers. Setup: the time-independent Schrödinger equation (eigenvalue problem): $(-\frac{\hbar^2}{2m}\Delta +V)\psi = E\psi$ (On ...
Rohan Didmishe's user avatar
6 votes
3 answers
696 views

How do I solve the following definite integral (preferably by an asymptotic method)?

$$ \int_{1}^{\infty} \frac{\sin^2 (\mu \sqrt{x^2 -1})}{(x+1)^{\frac{9}{2}} (x-1)^{\frac{3}{2}}} \,dx $$ Note: $\mu$ here is an extremely small constant. I have tried: Estimating the integral by ...
Abdullah's user avatar
4 votes
1 answer
102 views

Uniform decay of operator norm for smooth family of operators

Let $\mathscr{H}$ be a Hilbert space and let $\mathbb{R} \to B(\mathscr{H}), r \mapsto S_r$ be a continuous (or smooth) family of operators, where $B(\mathscr{H})$ is the space of bounded operators on ...
Constantin K's user avatar
7 votes
2 answers
230 views

Finding $\theta$ such that at least one eigenvalue of $A(\theta)$ is real

Is there a known method to find a set of $\theta$ such that at least one eigenvalue of $A(\theta)$ is purely real? Assume $A(\theta)$ is a real square matrix whose elements are linear functions of a ...
CWC's user avatar
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1 answer
184 views

Local differentiability of eigenvalues and eigenvectors of a real symmetric matrix

Let $A(x)\in\mathbb{R}^{n\times n}$ be a real symmetric matrix depending on the point $x\in\mathbb{R}^n$, where the eigenvalues are not necessarily simple. Can we say that for all $x$ there exists an ...
RS-Coop's user avatar
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1 answer
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Could variable be still function in x and y after performing Reynolds averaging over area

All, Let $S(x,y,t)$ be a variable function in $x$, $y$, and $t$. After performing Reynold averaging over area $\frac{1}{A}\int S(x,y,t) dA$, could $S$ still be a function in $x$, and $y$? Equations (1-...
Kernel's user avatar
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4 votes
1 answer
218 views

Asymptotics of integral representation of distribution

I initially posted this question at MSE (here), but I have gotten no response, so I figured I would ask it to this community. Background: I am studying the PDE $$\,\,\,\,\,\,\,\,\,\,\,\,i\partial_t \...
Dispersion's user avatar
2 votes
0 answers
135 views

Choosing the derivative of a flow

I am looking for something like the Franks' Lemma for flows. The celebrated Franks' Lemma states that: Let $f:M \rightarrow M$ be a $C^1$ diffeomorphism and $S=\{p_1,...,p_k\}$ be a finite set of ...
Ygor Arthur's user avatar
7 votes
1 answer
149 views

Is there an infinite combinatorics of common transseries expansions?

By now there is a rich understanding of generating functions in combinatorics, and the way that operations in power series are 'shadows' of richer constructions on combinatorial objects. This lifting ...
Robert Mastragostino's user avatar
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0 answers
42 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
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2 votes
0 answers
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Perturbation theory for $UV^*$ in singular value decomposition

There is a fair amount of research into perturbation theory for singular value decompositions (e.g. Liu et al's 'First-Order Perturbation Analysis of Singular Vectors in Singular Value Decomposition' ...
user7868's user avatar
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1 answer
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Bounding eigenvalue/eigenspace perturbations for hermitian matrices

Let $H$ be a Hermitian $n \times n$ matrix. Let $V$ be another such matrix. For real $t$, let us consider the one-parameter family $$ H(t) = H + t V$$ of Hermitian matrices. Kato's perturbation theory ...
Qualearn's user avatar
3 votes
0 answers
231 views

Is there a generalisation of this perturbation result about rank-one modifications of diagonal matrices?

In Theorem 1 of [1] we have the following result: Let $D$ be a real $n \times n$ diagonal matrix and consider the rank-one modification $C = D + \rho z z^T$, where $\rho > 0$ is a real scalar and $...
nanananille's user avatar
2 votes
2 answers
293 views

Solution of parabolic partial differential equation using singular perturbation method

Consider the following parabolic partial differential equation (PDE) \begin{align} \label{eq:42} \left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma}{r} \sin\psi \frac{\partial}{\partial \psi} + ...
GilbertDu's user avatar
0 votes
1 answer
227 views

Essential spectrum under perturbation

Given a Banach space $X$ and a bounded linear operator $T$ on $X$. It's well known that the essential spectrum of $T$ is invariant under additive compact perturbation. My question is about minimal ...
Malik Amine's user avatar
1 vote
1 answer
309 views

Eigenvalue perturbation under sparse perturbations

Let $A \in \{0,1\}^{n \times n}$ be an irreducible matrix whose entries are in $\{0,1\}$, and let $\lambda_1(A)$ be the eigenvalue with the largest magnitude. By Perron–Frobenius theorem, we know that ...
R. Davis's user avatar
5 votes
1 answer
118 views

Is Sun's spectral variation bound for normal matrices optimal?

In On the variation of the spectrum of a normal matrix, Sun proves the following result (Corollary 1.2): Let $A$ be an $n$-square normal matrix and $B$ an arbitrary $n$-square matrix. Then $$ \min_{\...
eepperly16's user avatar
3 votes
0 answers
143 views

Seeking a precedent – two-stage Gaussian integration?

Sometimes, by iteration, linear algebra can be used to solve non-linear equations. For example, consider the system $$Ax=a \qquad B(x)y=b(x), $$ where $a$ is a vector with scalar entries, $A$ is a ...
Dror Bar-Natan's user avatar
1 vote
2 answers
916 views

Lipschitz continuity of eigenvalues and eigenvectors of Hermitian matrices

It is well-known that the eigenvalues (in decreasing order) of a Hermitian matrix $A$ are Lipschitz continuous functions of $A$. Do there exist orthonormal eigenvectors that vary in a Lipschitz ...
Vamsi's user avatar
  • 3,323
3 votes
2 answers
250 views

Proof (or reference) about $\lambda_i(A+\epsilon e_je_j^*) = \lambda_i(A) + \epsilon |v_{i,j}|^2 + O(\epsilon^2).$

I'm looking for a proof (or a reference in a textbook) about the fact that $$ \lambda_i(A+\epsilon e_je_j^*) =_{\epsilon \to 0} \lambda_i(A) + \epsilon |v_{i,j}|^2 + O(\epsilon^2), $$ where $A$ is a ...
Michelle's user avatar
  • 161
0 votes
1 answer
156 views

Reference for action-angle coordinates [closed]

Does anyone know a good reference to start studying Action-Angle coordinates? Thank you in advance !
NSR's user avatar
  • 97
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1 answer
111 views

Perturbative approach starting from a probability distribution approximated form

I approximate a probability distribution $P_x(x)$ with a $P_x^{app}(x)$, such that $P_x(x)-P_x^{app}(x) = O(\epsilon)$ uniformly in x, where epsilon is a small positive quantity. Consider the generic ...
user1172131's user avatar
2 votes
0 answers
43 views

Stability in coefficients for the rescaled radiative tranport equation

One form of the radiative transport equation is as follows: $$ v\cdot \nabla_x u + \left(\epsilon \sigma_a(x) + \frac{1}{\epsilon}\sigma_s(x)\right) u - \frac{1}{\epsilon}\sigma_a(x)\int_{S^{n-1}} p(v,...
Phil's user avatar
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1 vote
1 answer
103 views

Sum of positive self-adjoint operator and an imaginary "potential": literature request

To keep things simple, let us consider the following: $L$ is a positive, unbounded S.A. operator on $L_2(\mathbb{R},f(x))$, where $f(x)$ is a Gaussian. Assume that we know the spectrum and ...
justin123's user avatar
0 votes
0 answers
89 views

Is a basis that almost diagonalizes a matrix 'close' to its eigenbasis?

Let $A\in\mathbb{R}^{n\times n}$ be diagonalisable (over $\mathbb{C}$) with pairwise distinct eigenvalues $\lambda_1,\ldots,\lambda_n\in\mathbb{C}$, and suppose that $$\tag{1}S^{-1}\cdot A\cdot S = \...
fsp-b's user avatar
  • 421
19 votes
1 answer
2k views

Why should I look at the resolvent formalism and think it is a useful tool for spectral theory?

Wikipedia calls resolvent formalism a useful tool for relating complex analysis to studying the spectra of a linear operator on a Banach space. Sure, I believe you because I've seen results that use ...
William Bell's user avatar
1 vote
0 answers
117 views

Perturbation in the equation $u_t=\epsilon Pu$, where $P$ is an elliptic partial differential operator

Let $Pu=\sum_{ij} \partial_j(a_{ij}(x) \partial_{i} u)$ be an elliptic operator. Consider the equation $$ (u=u_\epsilon)\\ \partial_t u=\epsilon Pu \text{ in } \mathbb R^+ \times \Omega,\\ u(0,x)=u_0(...
Ma Joad's user avatar
  • 1,591
0 votes
2 answers
242 views

Convergence of the eigenvector matrix for an analytic perturbation of a singular matrix

Let $A$ be an $n\times n$ matrix of all ones. Consider the analytic perturbation of $A$ as $$\tilde{A} = A + \epsilon H_1 + \epsilon^2 H_2 + \epsilon^3 H_3 + ... $$ All matrices are symmetric. Assume $...
Rajesh D's user avatar
  • 704
3 votes
0 answers
183 views

How to prove the following linearized operator is positive?

In $L^2(\mathbb{R}^d)$, let $Q$ be the solution to \begin{equation} -\Delta Q+\alpha^2 Q = |Q|^{2\sigma} Q, \end{equation} and $Q$ satisfies that it is positive, radial, and exponentially decaying (...
Tao's user avatar
  • 419
1 vote
1 answer
152 views

Perturbation of the value of a general-sum game at a equilibirium

Consider a general-sum game with $N$ players. Let $u_i(a_1, \ldots, a_N)\colon \prod_{i=1}^N A_i \rightarrow \mathbb{R} $ be the payoff of the player $i\in \{ 1, \ldots, N \}$ when each player takes ...
Steve's user avatar
  • 1,117
2 votes
0 answers
75 views

Two-variable singular perturbation analysis

I am having difficulty with a two-variable singular perturbation analysis on a set of ODE's. The key difficulty is also present in the following, embarrassingly simple problem: If $x\sim \mathcal{O}(...
tom's user avatar
  • 131
6 votes
0 answers
221 views

Poking into a Lie group with your finger

I consider this as a differential geometry problem. I have asked some of my classmates who are more interested in that, and also looked into some literature, but none of what I've found seems to help. ...
Student's user avatar
  • 5,008
9 votes
2 answers
874 views

Can we perturb a map $\mathbb{R}^n \to \mathbb{R}^n$ to have high rank?

Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth. Suppose that $df$ is invertible outside a set of Hausdorff dimension $\le n-1$, and ...
Asaf Shachar's user avatar
  • 6,611
12 votes
2 answers
967 views

A toy model in 0-d QFT

Questions For any positive integer $r$, compute $$(\frac{d}{dY})^r e^{(Y^2)}| _{Y=0}.$$ The answer should directly relates to a counting problem about Feynman diagrams. Is there a tutorial for how ...
Student's user avatar
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5 votes
1 answer
207 views

Can we perturb the Dirichlet boundary conditions to make harmonic maps locally invertible?

While analyzing a variational problem, I came to the following question: Let $\mathbb D^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball, and let $f: \mathbb D^n \to \mathbb{R}^n$ ...
Asaf Shachar's user avatar
  • 6,611
0 votes
0 answers
89 views

Coexistence of different solutions in a nonlinear matrix differential equation

I've faced a system of first-order nonlinear matrix differential equation, and I have tried to use perturbation method to approach the solutions. The differential equation has the form: \begin{...
sgyyhzd's user avatar
  • 43
7 votes
1 answer
482 views

Understanding Gillman's proof of the Chernoff bound for expander graphs

My question is about the proof of Claim 1 in this paper: Gillman (1993). At the end of the proof, the author says: The matrix product $U^\top\sqrt{D^{-1}}(P+(\mathrm{e}^x-1)B(0)-\mu I)\sqrt{D}U$, ...
Ella Sharakanski's user avatar
1 vote
0 answers
49 views

Perturbations of the eigen/singular directions

Let $A = U_A \Sigma_A V_A^\top$ and $B = U_B \Sigma_B V_B^\top$, and $A+B = U \Sigma V^\top$ be the respective singular vector decompositions. Is there some known relationship of the form $$\| U_A ...
John's user avatar
  • 21
2 votes
1 answer
722 views

$C^1$ perturbation of diffeomorphism is diffeomorphism?

if $f \in $ diff($M$), where $M$ is manifold, if $C^1$ perturbation $f_{\epsilon} $ of $f$ s.t. $||f_{\epsilon}-f||_{C^1} < \epsilon $. Can we prove $f_{\epsilon} \in $ diff($M$) if $\epsilon$ is ...
jason's user avatar
  • 553
1 vote
0 answers
102 views

stochastical stable

Given dynamic $f: S^1 \to S^1$ with Lebegue measure $dm$ on $S^1$. Assume it has unique SRB probability measure $\frac{d\mu_f}{dm} dm $. Given left shift space $([-\epsilon, \epsilon]^{\otimes \...
jason's user avatar
  • 553
2 votes
0 answers
299 views

Determinant of a rank r perturbation

In the following paper: Restricted Rank Modification of the Symmetric Eigenvalue Problem: Theoretical Considerations on page 79, Golub et al. have the following set of equations: $f(\lambda) = \...
tisPrimeTime's user avatar
4 votes
1 answer
460 views

Is the kernel of a Fredholm operator stable under perturbation?

This is a follow-up of this question. In a nutshell: Does the kernel of a bounded operator change "nicely" with the operator? Let $(X,\| \cdot \|)$ be an infinite-dimensional normed vector space. ...
Asaf Shachar's user avatar
  • 6,611
3 votes
1 answer
95 views

Bounds on $\|(P+\Delta)^n - P^n\|_F$ for stochastic matrices

Let us suppose that $P$ is a stochastic matrix (non-negative matrix with $P \mathbf{1} = \mathbf{1}$). Let $\Delta$ such that $P + \Delta$ is a stochastic matrix (which means $P + \Delta$ is non-...
mfrt's user avatar
  • 113
5 votes
1 answer
307 views

Analytic families of compact self-adjoint operators: eigenvalue extension

Suppose that $A(t), t \in \mathbb{R}$, is an analytic family of compact self-adjoint operators on a Hilbert space. The Kato-Rellich theorem says that every non-zero eigenvalue of $A(t)$ splits into ...
Brian Lins's user avatar
2 votes
1 answer
274 views

Eigenvalue Argument Perturbation

Given two square matrices $A$ and $B$. There are quite some results on the distance between the eigenvalues, e.g., $$ | \lambda_A - \lambda_B | \leq \| A - B \|_F, $$ where $A$ and $B$ are Hermitian ...
Jiro's user avatar
  • 909
5 votes
3 answers
757 views

Perturbation of a stochastic differential equation

Suppose we have the following two stochastic differential equations for $x_0$ and $x$ respectively \begin{align} dx_0 &= -k_0(t)(x_0-1)dt+\eta_0(t) x_0\,dB \tag1\\ dx &= -(k_0(t)+\epsilon ...
Hans's user avatar
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