Questions tagged [perturbation-theory]
The perturbation-theory tag has no usage guidance.
103
questions
19
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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 ...
18
votes
2
answers
4k
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Minimum off-diagonal elements of a matrix with fixed eigenvalues
I am an engineer working in radar research. I came accross a problem on which I cannot seem to find literature. I can ask it in two different ways. Perhaps depending on the reader, the alternative ...
13
votes
3
answers
3k
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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 ...
12
votes
2
answers
968
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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 ...
12
votes
2
answers
1k
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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 ...
11
votes
0
answers
243
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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 ...
10
votes
1
answer
1k
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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
$$x(t)=a_0+b_0e^{-t}+\epsilon(...
9
votes
2
answers
874
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Can we perturb a map $\mathbb{R}^n \to \mathbb{R}^n$ to have high rank?
Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth.
Suppose that $df$ is invertible outside a set of Hausdorff dimension $\le n-1$, and ...
9
votes
0
answers
474
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Rigorous results on the method of multiple scales
The method of multiple scales (Scholarpedia) is a technique used to obtain approximate solutions to differential equations, most commonly when some of the more standard approaches to perturbation ...
7
votes
2
answers
231
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Finding $\theta$ such that at least one eigenvalue of $A(\theta)$ is real
Is there a known method to find a set of $\theta$ such that at least one eigenvalue of $A(\theta)$ is purely real?
Assume $A(\theta)$ is a real square matrix whose elements are linear functions of a ...
7
votes
1
answer
150
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Is there an infinite combinatorics of common transseries expansions?
By now there is a rich understanding of generating functions in combinatorics, and the way that operations in power series are 'shadows' of richer constructions on combinatorial objects. This lifting ...
7
votes
1
answer
483
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Understanding Gillman's proof of the Chernoff bound for expander graphs
My question is about the proof of Claim 1 in this paper: Gillman (1993).
At the end of the proof, the author says:
The matrix product $U^\top\sqrt{D^{-1}}(P+(\mathrm{e}^x-1)B(0)-\mu I)\sqrt{D}U$, ...
6
votes
3
answers
698
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How do I solve the following definite integral (preferably by an asymptotic method)?
$$ \int_{1}^{\infty} \frac{\sin^2 (\mu \sqrt{x^2 -1})}{(x+1)^{\frac{9}{2}} (x-1)^{\frac{3}{2}}} \,dx $$
Note: $\mu$ here is an extremely small constant.
I have tried:
Estimating the integral by ...
6
votes
2
answers
405
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Can a perturbation of a matrix product always be represented as product of perturbations of its factor matrices?
Given $A=BC$ where $A\in\mathbb{R}^{m\times n}$ and for some $B\in\mathbb{R}^{m\times k}, C\in\mathbb{R}^{k\times n}$. We assume that $k>=\min(m,n)$ so that this decomposition always exists for any ...
6
votes
2
answers
264
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Gap-opening perturbations of the periodic Schrödinger operator
I am trying to understand this short paper and I am getting stuck right at the end.
Let $V(x)$ be $C^\infty$ and 1-periodic (that is, $V(x)=V(x+1)$).
We are considering the operator
$$A=-\dfrac{d^2}...
6
votes
1
answer
355
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Separating the spectrum of a Hermitian matrix
Given Hermitian matrix $A$, I would like to perturbate it so that its eigenvalues become well-separated.
Specifically, let $A$ be some Hermitian matrix, and let $G$ be a Gaussian matrix, with each
...
6
votes
0
answers
221
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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.
...
6
votes
0
answers
297
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Behavior of eigenspaces of adjacency matrices of random graphs (not via perturbation theory)
For the sake of discussion, let us say that we have the adjacency matrix $A$ of a graph, on $n$ nodes, from a stochastic block model with 2 blocks. Another name for this (usually used in computer ...
5
votes
3
answers
758
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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 ...
5
votes
1
answer
310
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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 ...
5
votes
1
answer
207
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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 $f: \mathbb D^n \to \mathbb{R}^n$ ...
5
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0
answers
133
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Series representation for unbounded perturbations of semigroup generators
Let $A$ generate an analytic $C_0$-semigroup on a Banach space $X$ and $B$ be a relatively compact perturbation, i.e., $B$ is compact as an operator from $D(A)$ (with the graph norm) to $X$. Then $A+B$...
5
votes
1
answer
119
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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 $$ \min_{\...
4
votes
1
answer
406
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Behaviour of eigenspaces of adjacency matrices after a single change to the graph
Say I know the eigenvalues and eigenvectors of an adjacency matrix of an unweighted graph. Can I say anything about the eigenvalues and eigenvectors of an adjacency matrix of a graph with one extra ...
4
votes
2
answers
1k
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Lower bounds for the singular values of submatrices of othogonal matrices
Let $A$ be an $m \times n$ matrix, $m\geq n$, and let $A=U\Sigma V^T$ be its singular value decomposition.
Let us partition $A$ as $A=(A_1|A_2)$, where $A_1$ is of size $m \times k$, and all columns ...
4
votes
1
answer
103
views
Uniform decay of operator norm for smooth family of operators
Let $\mathscr{H}$ be a Hilbert space and let $\mathbb{R} \to B(\mathscr{H}), r \mapsto S_r$ be a continuous (or smooth) family of operators, where $B(\mathscr{H})$ is the space of bounded operators on ...
4
votes
1
answer
218
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Asymptotics of integral representation of distribution
I initially posted this question at MSE (here), but I have gotten no response, so I figured I would ask it to this community.
Background: I am studying the PDE $$\,\,\,\,\,\,\,\,\,\,\,\,i\partial_t \...
4
votes
1
answer
161
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Sensitivity of the range of a matrix
The distance between two subspaces $\mathcal{U}$ and $\widetilde{\mathcal{U}}$ is classically defined as $d(\mathcal{U},\tilde{\mathcal{U}}):=\|P-\tilde{P}\|$, where $P$ and $\tilde{P}$ are orthogonal ...
4
votes
1
answer
461
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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.
...
4
votes
1
answer
356
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Distribution of the spectrum of a perturbed matrix
Let $A$ be an $n\times n$ Hermitian matrix,
with well-separated eigenvalues $\lambda_1 > \lambda_2 ... > \lambda_n$,
with $|\lambda_i-\lambda_j|>\epsilon$, for all $i \neq j$.
Let $G$ be a ...
4
votes
0
answers
80
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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}^{...
3
votes
1
answer
321
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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 ...
3
votes
2
answers
252
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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 ...
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-...
3
votes
1
answer
614
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 $...
3
votes
0
answers
238
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 $...
3
votes
0
answers
144
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 ...
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 (...
3
votes
0
answers
47
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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 ...
3
votes
0
answers
182
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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 ...
3
votes
0
answers
379
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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 ...
3
votes
1
answer
530
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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 ...
2
votes
1
answer
736
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 ...
2
votes
3
answers
2k
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Analytic perturbation of the eigenvalues/eigenvectors of non-Hermitian matrix
Consider a matrix function $A(x)$, analytically depending on single parameter $x$.
Consider the eigenvalue/eigenvector pair of $A(0)$, namely $\lambda(0)$ and $w(0)$.
The question is whether we can ...
2
votes
2
answers
394
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
votes
1
answer
630
views
Perturbation theory of eigenvalues - Effects of degeneracy/ multiplicity
In Quantum mechanics Schrödinger's perturbation theory is very important (see Wikipedia) which deals with perturbation of the discrete spectrum of a self-adjoint operator.
Where can I find a ...
2
votes
1
answer
84
views
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 ...
2
votes
1
answer
1k
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 ...
2
votes
1
answer
250
views
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 ...
2
votes
1
answer
700
views
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\...