Questions tagged [perturbation-theory]

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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 ...
xmonetx's user avatar
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1 vote
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76 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 ...
user3037237's user avatar
2 votes
0 answers
55 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)\\ &...
sleeve chen's user avatar
3 votes
0 answers
47 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 ...
gradstudent's user avatar
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3 votes
0 answers
182 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 ...
Appliqué's user avatar
  • 1,269
12 votes
2 answers
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 ...
thedude's user avatar
  • 1,417
3 votes
1 answer
321 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 ...
mystupid_acct's user avatar
1 vote
1 answer
148 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 & \\ \...
vkonton's user avatar
  • 175
1 vote
0 answers
74 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 ...
Siam's user avatar
  • 11
1 vote
0 answers
110 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 ...
valle's user avatar
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4 votes
0 answers
80 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}^{...
Serghei's user avatar
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6 votes
2 answers
405 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 ...
jayki's user avatar
  • 135
4 votes
2 answers
1k 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 ...
cangrejo's user avatar
  • 161
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0 answers
189 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 ...
JJUser's user avatar
  • 11
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 ...
Bill J's user avatar
  • 65
0 votes
1 answer
550 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 ...
Michel's user avatar
  • 3
2 votes
0 answers
173 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 ...
Daniel86's user avatar
  • 215
11 votes
0 answers
243 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 ...
Gordon Royle's user avatar
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1 vote
1 answer
165 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-...
bittertea's user avatar
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 $...
eddard's user avatar
  • 81
0 votes
0 answers
66 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 ...
Keith McClary's user avatar
6 votes
2 answers
264 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}...
Darren Ong's user avatar
3 votes
1 answer
530 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 ...
R S's user avatar
  • 985
4 votes
1 answer
405 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 ...
user91322's user avatar
1 vote
1 answer
1k 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 ...
Val K's user avatar
  • 355
1 vote
2 answers
392 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. ...
Val K's user avatar
  • 355
0 votes
1 answer
105 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 ...
Drn004's user avatar
  • 103
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 $x^*$ ...
rajatsen91's user avatar
2 votes
1 answer
142 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 ...
Val K's user avatar
  • 355
10 votes
1 answer
1k 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(...
Conifold's user avatar
  • 1,599
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 ...
Guest's user avatar
  • 123
3 votes
0 answers
379 views

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 ...
user82891's user avatar
1 vote
0 answers
41 views

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 ...
John Taylor's user avatar
2 votes
0 answers
70 views

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 ...
QuantumTheory's user avatar
1 vote
0 answers
146 views

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, ...
shahulhameed's user avatar
1 vote
0 answers
295 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 ...
user62650's user avatar
0 votes
1 answer
248 views

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}+\...
FraSchelle's user avatar
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\...
Helmholz's user avatar
  • 131
0 votes
1 answer
203 views

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\...
timur's user avatar
  • 3,292
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 ...
Chandler's user avatar
4 votes
1 answer
161 views

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 ...
Federico Poloni's user avatar
1 vote
0 answers
122 views

Inverse of matrix of generalised harmonic numbers

For $s=0,1,\dots$ and $n=1,2,\dots$, denote $r_{n,s}=\sum_{k=1}^n k^s$. It is well-known that $r_{n,s}$ are polynomials in $n$ with leading term $\frac{1}{s+1}n^{s+1}$. Let $R_{n,s}$ be the $(s+1)\...
dima's user avatar
  • 949
2 votes
0 answers
732 views

Distributions of eigenvalues for matrix normal distribution: related references

I am interested in the distribution of the eigenvalues of matrices that are sampled from the matrix normal distribution. I am sampling from $p(X \mid M,U,V)$ and let's assume that I know the ...
jkt's user avatar
  • 169
1 vote
2 answers
382 views

Is Rellich's function valued theorem valid for a rank defficient function valued matrix?

Theorem (Rellich). Let $\boldsymbol{A}(t) : \mathbb{R}\rightarrow\mathbb{C}^{n \times n}$ be a Hermitian matrix function that depends on $t$ analytically. (i) The $n$ roots of the characteristic ...
trienko's user avatar
  • 33
6 votes
1 answer
355 views

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 ...
Lior Eldar's user avatar
2 votes
3 answers
2k views

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 ...
bcp's user avatar
  • 165
4 votes
1 answer
356 views

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 ...
Lior Eldar's user avatar
5 votes
0 answers
133 views

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 $A+B$...
Delio Mugnolo's user avatar
6 votes
0 answers
297 views

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 ...
passerby51's user avatar
  • 1,639
1 vote
0 answers
104 views

On generalization of Wigner semi circle

I want to analyse noise model for a matrix M whose entries are not real numbers. The matrix is a collection of N permutation matrices of size nxn i.e, M is NnxNn. Because its a collection of ...
Deepti Pachauri's user avatar