Questions tagged [perturbation-theory]
The perturbation-theory tag has no usage guidance.
103
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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 ...
1
vote
0
answers
76
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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 ...
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)\\
&...
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 ...
12
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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 ...
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 ...
1
vote
1
answer
148
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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 & \\
\...
1
vote
0
answers
74
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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 ...
1
vote
0
answers
110
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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 ...
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}^{...
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 ...
4
votes
2
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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 ...
1
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0
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189
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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 ...
2
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1
answer
1k
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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 ...
0
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1
answer
550
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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 ...
2
votes
0
answers
173
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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 ...
11
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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 ...
1
vote
1
answer
165
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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-...
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 $...
0
votes
0
answers
66
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$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 ...
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
$$A=-\dfrac{d^2}...
3
votes
1
answer
530
views
Second-order perturbation expansion for singular value decomposition
Let $A = U\Sigma V^T$ be the singular value decomposition (SVD) of a $n\times m$ matrix $A$. Let $\tilde{A} = A + \epsilon P$ be a perburbation of $A$. It is possible, using tools from Matrix ...
4
votes
1
answer
405
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Behaviour of eigenspaces of adjacency matrices after a single change to the graph
Say I know the eigenvalues and eigenvectors of an adjacency matrix of an unweighted graph. Can I say anything about the eigenvalues and eigenvectors of an adjacency matrix of a graph with one extra ...
1
vote
1
answer
1k
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Spectral radius's relation with row sum
Let $A$ be a non-negative $N \times N$ square matrix with $a_{i,i}=0, 1 \leq i \leq N$. Also, let $r_i$ be the $i$-th row sum of $A$.
I know that $\rho(A)$, the spectral radius of $A$, is bounded as ...
1
vote
2
answers
392
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Spectral radius of a non-negative matrix after moving and replicating an element
Let $A$ be a non-negative square matrix and its spectral radius (i.e., it's largest eigenvalue) be $\rho(A)$. I need to do the following operation to $A$ and compare the resulting spectral radii.
...
0
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1
answer
105
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Eigenvalue-related statements [closed]
(I understand this question might not be appropriate for this website, but it has been asked on MathStackexchange and did not receive any replies even with a bounty)
How can I prove that the ...
2
votes
2
answers
394
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Perturbation of Linear Programs
Consider the linear program,
$$\begin{array}{ll} \text{maximize} & c^T x\\ \text{subject to} & Ax \leq b\\
& x \geq 0\end{array}$$
I want to study the sensitivity of the optimal $x^*$ ...
2
votes
1
answer
142
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Spectral radius of perturbed bipartite graphs
I am looking into how perturbation(s) on a bipartite graph affect its spectrum (specifically its spectral radius or largest eigenvalue). Actually, I'm not exactly looking into bipartite but my ...
10
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1
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How to eliminate secular terms for perturbed non-oscillatory equations?
Even in a linear second order equation like $x''+x'+\epsilon x=0$ the standard asymptotic expansion has a secular term already in the first order of $\epsilon$, namely
$$x(t)=a_0+b_0e^{-t}+\epsilon(...
2
votes
1
answer
250
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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 ...
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 ...
1
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0
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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 ...
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 ...
1
vote
0
answers
146
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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, ...
1
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0
answers
295
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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 ...
0
votes
1
answer
248
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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}+\...
2
votes
1
answer
700
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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\...
0
votes
1
answer
203
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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\...
2
votes
1
answer
630
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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 ...
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 ...
1
vote
0
answers
122
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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)\...
2
votes
0
answers
732
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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 ...
1
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2
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382
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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 ...
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
...
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 ...
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 ...
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 $A+B$...
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 ...
1
vote
0
answers
104
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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 ...