# Questions tagged [perturbation-theory]

The perturbation-theory tag has no usage guidance.

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### $C^1$ 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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### 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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55 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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166 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.
...

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### 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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### 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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190 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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305 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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864 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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### sensitive perturbation approximation

I was reading paper which associated with perturbation approximation. paper1 paper2.
In paper1:
$\bar{R}=R+\epsilon C$, first order: when $\Lambda_1\gg\Lambda_2$, $\Delta\Lambda_{max}=\frac{\vec{v}^...

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### 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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### 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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### 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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### 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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### 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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138 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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57 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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55 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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75 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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### Continuity/differentiability of eigenvectors corresponding to semisimple eigenvalues

From my reading and intuition, I'm pretty sure that the following is true:
The eigenvectors corresponding to semisimple eigenvalues (i.e. algebraic multiplicity = geometric multiplicity) of the ...

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224 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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### 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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### 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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295 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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### 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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### 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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### 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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141 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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### 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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### $H$ 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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### 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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271 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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### Linear systems of equations with singular coefficient matrix [closed]

Consider a consistent system of linear equations $Ax=b$. Let's assume for simplicity that $A$ is square $n \times n$. We are looking for an effectively computable approximate solution $\hat{x}$.

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### 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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294 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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194 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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### 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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### 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 $x^*$ ...

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### 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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### 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(...

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### A clarification regarding analytic perturbation of metrics and Laplacian

This question is in reference to the following Mathoverflow question and the accepted answer to it. It seems to me that it is taken for granted that if the metric $g_t$ perturbs real analytically in ...

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### Smooth perturbation of a positive self-adjoint operator with compact resolvent

Consider a one-parameter family $A_t$ of unbounded positive self-adjoint operators with discrete spectrum (for example, one can consider a one-parameter family of Laplacians on a compact Riemannian ...

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### How to treat equation with alternating square of frequency?

Let's have equation
$$
\tag 1 \frac{d^{2}y(t)}{dt^{2}} +\omega^{2}(t)y(t) = 0, \quad t \in (t_{\text{in}}, \infty)
$$
Here
$$
\omega^{2}(t) = A(t) - B(t)cos(2t),
$$
and functions $A(t), B(t)$ have ...

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### Error bounds for eigenvalue expansion of the Mathieu equation

The Mathieu equation is an important eigenvalue problem in Mathematical Physics that is completely understood in its properties, although there is no "direct way" of expressing eigenvalues and ...

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### 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, ...

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249 views

### Perturbation method of a boundary value problem

Let $u(x, \epsilon, \theta)$ be the solution of $$u''+(\epsilon \cos(x)+\theta-u)u=0$$ with boundary conditions $u'(0)=0$ and $u'(\pi)=0$. Here $\theta\in [0, 1]$.
I tried to put the solution in ...

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### A hyperbolic partial differential equation (wave-like) with variable-dependent coefficient and possibly singular in one variable

First, I beg your pardon since the title of the question is a bit confusing I guess. I'm working on a physical equation of the wave-like form. Explicitly, it reads
$$\left[\left(\cos\phi\partial_{z}+\...

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### What is the relation between the eigenvectors of a sample covariance matrix and those of the true covariance matrix?

As is known, the covariance matrix of a set of random vectors $\{\mathbf{x}_i\}_{i=1}^N$ can be estimated by their sample covariance matrix:
$\mathbf{\hat R}:=\frac{1}{N}\sum_{n=1}^N\mathbf{x}_n\...

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### Strongly convergent series of bounded self-adjoint operators

Let $A_n$ and $A$ be bounded self-adjoint operators in a Hilbert space, such that $A_n\to A$ strongly. Then it is well known that $(z-A_n)^{-1}\to(z-A)^{-1}$ strongly for each $z\in\mathbb{C}\setminus\...