Questions tagged [perturbation-theory]

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31 views

Floquet theory and Poincaré theorem on the continuation of periodic orbit

I read about the Floquet theory and a theorem that it named Poincaré's theorem of the continuation of periodic orbit. Poincaré's Theorem: Consider a dynamical system depending on the parameter $\...
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28 views

Perturbation analysis and sensitivity of eigenvector matrix product with specific perturbation

In my research in applied linear algebra and probability (Wiener filtering) I have come across this rather interesting problem: For a matrix $ U $ we denote by $ U_k $ the matrix formed by taking ...
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40 views

Iterative method to find the derivative of eigenvectors of a large matrix

Given a large real symmetric matrix, $\mathbf{A(m)}\in\mathbb{R}^{N\times N}$, which elements depend on a parameters vector, $\mathbf{m}\in\mathbb{R}^m$. The matrix elements cannot be stored ...
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1answer
49 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 ...
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40 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}(...
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189 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. ...
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2answers
833 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 ...
12
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2answers
761 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 ...
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1answer
184 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 $f: \mathbb D^n \to \mathbb{R}^n$ ...
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45 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{...
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1answer
370 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 $U^\top\sqrt{D^{-1}}(P+(\mathrm{e}^x-1)B(0)-\mu I)\sqrt{D}U$, ...
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33 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 ...
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1answer
233 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 ...
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89 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 \...
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98 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) = \...
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1answer
258 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. ...
3
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1answer
83 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-...
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137 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 ...
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1answer
217 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 ...
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3answers
404 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 ...
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1k views

Differentiability of Eigenvalues - Perturbation Theory

first, I have a general question. In perturbation theory, I saw perturbations in eigenvalues and eigenvectors of square, non-symmetric matrices and the calculations were all right but no one ever ...
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67 views

Showing a modified system of quadratic equations is stable

I have and $n$ dimensional dynamical system, given by $\dot{x} = M D(x) P x - \frac{c}{2}x$ $P$ is a full rank $n \times n$ matrix, with $p_{ij} \in [0,c]$, such that $p_{ij}=c-p_{ji}$ for some ...
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48 views

Region of attraction of simple ODE with perturbation

Consider the following simplest example: $$\dot{x} = x(x-1)(x+1)$$ $[-1,1]$ is the ROA. Now consider the two dimensional case: \begin{equation} \begin{aligned} &\dot{x} = x(x-1)(x+1)\\ &...
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43 views

Questions on “The condition number of a randomly perturbed matrix”

This question is about the two vectors $w'$ and $y$ that are necessary for the argument in section $7$ (page 6) of this paper by Terence Tao and Van Vu, https://arxiv.org/abs/math/0703307 (that ...
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78 views

Asymptotic stability of eigenvalues by compact perturbations

I need some references concerning the asymptotic stability of eigenvalues by compact perturbations. In [T. Kato, Perturbation theory for linear operators] there are some results concerning stability ...
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2answers
1k views

Eigenvalue perturbation theory via Feynman diagrams

Suppose I have a matrix given by a sum $$A=D+\epsilon B$$ where $D$ is diagonal and $\epsilon$ is small, and I want the eigenvalues of $A$ as a power series in $\epsilon$. The first two orders in ...
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1answer
190 views

Inverting (via Taylor expansion) a sum of (rank-deficient) skew-symmetric matrix and (rank-deficient) Diagonal matrix

I have the following problem: A matrix $C\in \mathbb{R}^{2N}$, where $C=\epsilon A+D$ $\epsilon A=(C-C')/2$ is skew symmetric with "block" anti-diagonal structure of size 4. $ D=(C+C')/2$ (Diagonal ...
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1answer
62 views

Solving Linear System with Noisy Input

I have the following triangular system \begin{equation} \begin{pmatrix} 1 & & & & \\ \mu_1 & 2 & & & \\ \mu_2 & \mu_1 & 3 & \\ \...
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66 views

Time dependent Hamiltonians

I'm studying time dependent perturbation theory on Reed-Simon book "Method of modern mathematical physics, II". If one considers an Hamiltonian of the form $$H(t)=H_0+V(t)$$ the corresponding formal ...
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76 views

Perturbation theory for the homogeneous Fredholm integral equation of the second kind

Is there any accessible treatment of perturbation theory for homogeneous Fredholm integrals of the second kind? Specifically, suppose I have a kernel $K(x,y) = K_0(x,y) + \eta(x,y)$ that is obtained ...
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76 views

Perturbation of a rank-restricted product of matrices

I formulated a statement, which is hopefully true (at least I'm not knowledgeable enough to see a reason for it not to be). However, I'm struggling to come up with a proof. Let $W_i \in \mathbb{R}^{...
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2answers
269 views

Can a perturbation of a matrix product always be represented as product of perturbations of its factor matrices?

Given $A=BC$ where $A\in\mathbb{R}^{m\times n}$ and for some $B\in\mathbb{R}^{m\times k}, C\in\mathbb{R}^{k\times n}$. We assume that $k>=\min(m,n)$ so that this decomposition always exists for any ...
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2answers
551 views

Lower bounds for the singular values of submatrices of othogonal matrices

Let $A$ be an $m \times n$ matrix, $m\geq n$, and let $A=U\Sigma V^T$ be its singular value decomposition. Let us partition $A$ as $A=(A_1|A_2)$, where $A_1$ is of size $m \times k$, and all columns ...
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100 views

Additive perturbation bounds on the eigenvectors of a Hermitian matrix

I am reading this paper: http://society.math.ntu.edu.tw/~journal/tjm/V16N1/TJM-258.pdf where the authors find additive perturbation bounds on the matrix of the eigenvectors of a Hermitian matrix. I ...
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1answer
485 views

Exact first order perturbation expansion of matrix determinant [closed]

Let $I$ be an $n\times n$ identity matrix, $B$ be an $n\times n$ matrix with all the elements tending to zero. If we can expression $det(I+B)= 1 + f(B) + o(f^2(B))$, as all the elements of $B$ go to ...
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1answer
515 views

Perturbation theory for matrices

I encountered the following problem. Since this is somewhat not related to what I normally do, I wanted to know what the best estimates in this field are. Let $A \in \mathbb{R}^{n \times n}$ be a ...
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0answers
141 views

Size of Jordan blocks under random perturbations

Let $A \in \mathbb{C}^{n \times n}$ be some (fixed) matrix with eigenvalues $\lambda_{1},\ldots,\lambda_{n}$. Let $E$ be some random, small-normed, perturbation such that $\tilde{A} = A+E$ has ...
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186 views

Is my matrix perturbation analysis legitimate?

I am not a matrix theorist, or numerical linear algebra expert, but I have a problem and my proposed solution leads me to a question that I cannot answer. I can give more details, but the gist is ...
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1answer
147 views

Find a matrix and its inverse satisfying lower and upper bounds

I reduced a problem of matrix completion to the problem find $A,B$ such that $AB=I$ $A_{min}\leq A \leq A_{max}$ $B_{min}\leq B \leq B_{max}$ One possible approach would be to just minimize $\|AB-...
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0answers
210 views

perturbed vs. unperturbed Hamiltonian system

Let's take a time-periodic Hamiltonian $H(t,x,y)$ on $\mathbb{R}^2$ and apply an arbitrarily small time-independent perturbation to $H$ via $$ \tilde H (t,x,y) = H(t,x,y) + \epsilon V(x,y), $$ where $...
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57 views

$H$ self-adjoint with mass gap, $P≥0,Ω∈D(P),H+λP$ self-adjoint $⟹$ for $λ$ small, $H+λP$ has gap?

Suppose $H$ is a self-adjoint operator on a Hilbert space having a simple isolated least eigenvalue $0$ with gap $1$ ( $H\Omega = 0$, $\Vert \Omega\Vert = 1$ ), $P$ is a non-negative symmetric ...
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2answers
216 views

Gap-opening perturbations of the periodic Schrödinger operator

I am trying to understand this short paper and I am getting stuck right at the end. Let $V(x)$ be $C^\infty$ and 1-periodic (that is, $V(x)=V(x+1)$). We are considering the operator $$A=-\dfrac{d^2}...
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1answer
331 views

Second-order perturbation expansion for singular value decomposition

Let $A = U\Sigma V^T$ be the singular value decomposition (SVD) of a $n\times m$ matrix $A$. Let $\tilde{A} = A + \epsilon P$ be a perburbation of $A$. It is possible, using tools from Matrix ...
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1answer
250 views

Behaviour of eigenspaces of adjacency matrices after a single change to the graph

Say I know the eigenvalues and eigenvectors of an adjacency matrix of an unweighted graph. Can I say anything about the eigenvalues and eigenvectors of an adjacency matrix of a graph with one extra ...
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1answer
481 views

Spectral radius's relation with row sum

Let $A$ be a non-negative $N \times N$ square matrix with $a_{i,i}=0, 1 \leq i \leq N$. Also, let $r_i$ be the $i$-th row sum of $A$. I know that $\rho(A)$, the spectral radius of $A$, is bounded as ...
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2answers
252 views

Spectral radius of a non-negative matrix after moving and replicating an element

Let $A$ be a non-negative square matrix and its spectral radius (i.e., it's largest eigenvalue) be $\rho(A)$. I need to do the following operation to $A$ and compare the resulting spectral radii. ...
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1answer
99 views

Eigenvalue-related statements [closed]

(I understand this question might not be appropriate for this website, but it has been asked on MathStackexchange and did not receive any replies even with a bounty) How can I prove that the ...
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2answers
281 views

Perturbation of Linear Programs

Consider the linear program, $$\begin{array}{ll} \text{maximize} & c^T x\\ \text{subject to} & Ax \leq b\\ & x \geq 0\end{array}$$ I want to study the sensitivity of the optimal $x^*$ ...
2
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1answer
112 views

Spectral radius of perturbed bipartite graphs

I am looking into how perturbation(s) on a bipartite graph affect its spectrum (specifically its spectral radius or largest eigenvalue). Actually, I'm not exactly looking into bipartite but my ...
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1answer
398 views

How to eliminate secular terms for perturbed non-oscillatory equations?

Even in a linear second order equation like $x''+x'+\epsilon x=0$ the standard asymptotic expansion has a secular term already in the first order of $\epsilon$, namely $$x(t)=a_0+b_0e^{-t}+\epsilon(...