Questions tagged [perturbation-theory]
The perturbation-theory tag has no summary.
119 questions
6
votes
2
answers
488
views
A question related to matrix inverse diagonal zero property
$\DeclareMathOperator\supp{supp}$Let a symmetric nonsingular matrix $A \in \mathbb{R}^{2n \times 2n} $ have the following block form
$$ A = \begin{bmatrix}
X & D \\
D^{\top} ...
- 61
0
votes
0
answers
60
views
analyzing the sensitivity of two matrix expressions
I'm working on analyzing the sensitivity of two matrix expressions. I'd like to formally show that one is more sensitive to perturbations in a covariance matrix C than the other.
We are given:
$$\...
- 11
5
votes
1
answer
254
views
Independence of parameter for eigenvalues of periodic family of tridiagonal matrices
Consider the family of matrices $C(\ell,\theta)\in \mathbb{R}^{2\ell+2\times2\ell+2}$, where $\ell\in\mathbb{N}$ and $\theta\in\mathbb{R}$ given by
\begin{equation*}
C(\ell,\theta)=\begin{pmatrix}
...
- 153
0
votes
0
answers
84
views
How to find perturbation to a function defined by a perturbed implicit relation
Statement:
Suppose we have a relation $F(x,y,z)=0$ from which we can explicit find a function $z=f(x,y)$. Now suppose we have a new (perturbed) relation
$$F(x,y,z)+hG(x,y,z)=0$$
where $F$ is a known ...
- 562
2
votes
1
answer
144
views
Choosing eigenvectors continuously for positive-semidefinite matrix function of rank one
Consider a real-analytic, rank-one, matrix-valued function $M(t)\geq 0$ of single real variable $t$. Can one choose a symmetric factorization $M(t) = z(t) z^T(t)$ (where $z(t)$ is an eigenvector with ...
- 23
2
votes
0
answers
112
views
The relation between capacity and the Poincaré inequality
Let $N > 2$ and let $\omega, \, \Omega \subset \mathbb{R}^N$ be domains containing the origin. Define $\varepsilon \omega := \{\varepsilon x : x \in \omega\}$ for $\varepsilon > 0$. I am ...
2
votes
1
answer
194
views
Bauer-Fike theorem
I have a doubt about the interpretation of the Bauer-Fike theorem. It states that:
Given $ A \in \mathbb{C}^{N \times N} $ diagonalizable matrix ($ A = S D S^{−1} $ and $ D $ diagonal matrix having ...
- 23
0
votes
0
answers
76
views
Existence of a perturbation preserving positivity and spectral bounds for a positive linear map on symmetric matrices
Let $ T: \operatorname{Sym}^{d \times d} \to \operatorname{Sym}^{d \times d} $ be a linear map that is positive, meaning that if $ \mathbf{X} \in \operatorname{Sym}^{d \times d} $ is positive ...
- 123
-5
votes
1
answer
154
views
Why is the second order correction to energy zero for a fully degenerate eigensystem? [closed]
Consider the system given by,
$$ H|n\rangle = E|n\rangle$$
where:
$H$ is the hamiltonian.
$|n\rangle$ is the eigenstate.
$E$ is the energy of the eigenstate.
Using degenerate perturbation theory and ...
- 45
0
votes
1
answer
227
views
Can the derivative of eigenvectors with respect to its components be taken as zero if all eigenvalues are equal?
I want to ask a couple of follow up questions to the question answered on the thread "Derivative of eigenvectors of a matrix with respect to its components".
I noticed that in the accepted ...
- 45
3
votes
1
answer
179
views
Error bound for MonteCarlo estimate of elements in Gram-Matrix
Suppose I have a $n\times n$-symmetric positive-definite matrix $A$ with elements:
\begin{align}
[A]_{ij}=\int_{\Omega}f_i(x)f_j(x) \, dx, \quad i,j=1,\ldots,n
\end{align}
where $\Omega\subset \mathbb{...
- 103
2
votes
0
answers
109
views
Approximate solutions to $x''(t)=-cx + f(t)x$
I recently studied a problem which involved two particles joined by a harmonic spring moving in a potential and through some manipulation, I obtained the equation
$x''(t) = -\omega^2x + f(t)x$,
where $...
- 3,808
4
votes
1
answer
274
views
First derivative of $f(A) = \frac{1}{\lambda_{\min}(A)}$ for perturbed matrix
I am working with the matrix function
$$
f(A) = \frac{1}{\lambda_{\min}(A)},
$$ where $A \in \mathbb{R}^{n \times n}$ is a positive definite matrix and $\lambda_{\min}(A)$ is its smallest eigenvalue. ...
- 91
4
votes
0
answers
164
views
Stability of first eigenfunction of Laplace-Beltrami in spherical caps
Let us denote $x \in \mathbb{R}^n$ by $(x',x_n)$, where $x' \in \mathbb{R}^{n-1}$.
Let $\Omega_L := \{x : |x| = 1, x_n > L|x'|\} \subset \mathbb{S}^{n-1}.$
Then, we consider $\phi_L$ to be the ...
1
vote
0
answers
175
views
Solve coupled ODEs analytically in the limit of a small parameter
I have the following set of coupled second order non-linear ODEs :
$$ x^2 a''(x) + x a'(x) - \Big(\frac{1}{\epsilon^2}\Big)b^2(x) a(x) = 0 \\
x b''(x) - b'(x) - 2x b(x)a^2(x) = 0$$
with boundary ...
- 11
3
votes
1
answer
210
views
Low rank perturbation of non-Hermitian $A$, where all eigenvalues are real
Suppose $A,E$ are Hermitian $(n \times n)$-matrices and $E$ is of low rank. There are well known results bounding the difference in spectra of $A$ and $A+E$. For example the Eigenvalue Interlacing ...
- 1,389
2
votes
1
answer
267
views
Lipschitz continuity of eigenprojections
This question has the same flavor of this and this questions, but asks for something stronger.
Assume that
$A$ is a symmetric $n \times n$ matrix,
$H$ is a $n \times n$ perturbation matrix.
Moreover ...
- 39
1
vote
0
answers
84
views
Bounds on the spectral radius of a perturbed directed graph
Suppose $(G_n)$ is a sequence of strongly connected directed graphs (without multiple edges) with $G_n$ having $n$ edges such that the adjacency matrix $A_n$ of $G_n$ is primitive, and let $(G_n’)$ be ...
- 11
1
vote
0
answers
68
views
How to get perturbation bounds of singular vectors
Let an adjacency matrix $A={A^\top}\in {\mathbb{R}^{n \times n}}$ (a binary matrix) of a simple undirected graph and its degree matrix $D$ be given.
When adding $Q$ edges into the graph, which is ...
- 11
2
votes
0
answers
160
views
Linear elliptic problems: Are gradient estimates preserved after perturbation?
(This question is a duplicate from here)
We start with the linear elliptic PDE
$$
-\operatorname{div}(A\nabla u)=f \quad\text{in}\ \Omega,\\
u=0 \quad\text{on}\ \partial\Omega
$$
We assume that $\...
- 141
1
vote
0
answers
145
views
Schrödinger equation approximation – continuity of eigenvalues with respect to potential
The question has been crossposted from Stackexchange after receiving no answers.
Setup: the time-independent Schrödinger equation (eigenvalue problem):
$(-\frac{\hbar^2}{2m}\Delta +V)\psi = E\psi$
(On ...
- 173
6
votes
3
answers
867
views
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 ...
- 69
4
votes
1
answer
147
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 ...
- 429
7
votes
2
answers
286
views
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 ...
- 451
0
votes
1
answer
315
views
Local differentiability of eigenvalues and eigenvectors of a real symmetric matrix
Let $A(x)\in\mathbb{R}^{n\times n}$ be a real symmetric matrix depending on the point $x\in\mathbb{R}^n$, where the eigenvalues are not necessarily simple. Can we say that for all $x$ there exists an ...
- 39
0
votes
1
answer
160
views
Could variable be still function in x and y after performing Reynolds averaging over area
All,
Let $S(x,y,t)$ be a variable function in $x$, $y$, and $t$. After
performing Reynold averaging over area $\frac{1}{A}\int S(x,y,t) dA$, could $S$ still be a function in $x$, and $y$?
Equations (1-...
- 101
4
votes
1
answer
274
views
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 $$\,\,\,\,\,\,\,\,\,\,\,\,i\partial_t \...
- 946
2
votes
0
answers
173
views
Choosing the derivative of a flow
I am looking for something like the Franks' Lemma for flows. The celebrated Franks' Lemma states that: Let $f:M \rightarrow M$ be a $C^1$ diffeomorphism and $S=\{p_1,...,p_k\}$ be a finite set of ...
- 121
7
votes
1
answer
203
views
Is there an infinite combinatorics of common transseries expansions?
By now there is a rich understanding of generating functions in combinatorics, and the way that operations in power series are 'shadows' of richer constructions on combinatorial objects. This lifting ...
1
vote
0
answers
94
views
Small perturbation to a commuting family of hermitian matrices will hurt the nice properties?
Let $A_1, \dotsc A_N$ be a collection of finite Hermitian matrices that commute with one another and all have the matrix $2$-norm as $1$. Here $N$ is large but fixed.
Then, they are simultaneously ...
- 3,745
2
votes
0
answers
251
views
Perturbation theory for $UV^*$ in singular value decomposition
There is a fair amount of research into perturbation theory for singular value decompositions (e.g. Liu et al's 'First-Order Perturbation Analysis of Singular Vectors
in Singular Value Decomposition' ...
- 665
2
votes
1
answer
152
views
Bounding eigenvalue/eigenspace perturbations for hermitian matrices
Let $H$ be a Hermitian $n \times n$ matrix. Let $V$ be another such matrix.
For real $t$, let us consider the one-parameter family
$$ H(t) = H + t V$$
of Hermitian matrices.
Kato's perturbation theory ...
- 143
4
votes
0
answers
553
views
Is there a generalisation of this perturbation result about rank-one modifications of diagonal matrices?
In Theorem 1 of [1] we have the following result: Let $D$ be a real $n \times n$ diagonal matrix and consider the rank-one modification $C = D + \rho z z^T$, where $\rho > 0$ is a real scalar and $...
- 141
2
votes
2
answers
475
views
Solution of parabolic partial differential equation using singular perturbation method
Consider the following parabolic partial differential equation (PDE)
\begin{align}
\label{eq:42}
\left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma}{r} \sin\psi \frac{\partial}{\partial \psi} + ...
- 91
0
votes
1
answer
340
views
Essential spectrum under perturbation
Given a Banach space $X$ and a bounded linear operator $T$ on $X$.
It's well known that the essential spectrum of $T$ is invariant under additive compact perturbation.
My question is about minimal ...
- 299
3
votes
1
answer
522
views
Eigenvalue perturbation under sparse perturbations
Let $A \in \{0,1\}^{n \times n}$ be an irreducible matrix whose entries are in $\{0,1\}$, and let $\lambda_1(A)$ be the eigenvalue with the largest magnitude. By Perron–Frobenius theorem, we know that ...
- 31
5
votes
1
answer
159
views
Is Sun's spectral variation bound for normal matrices optimal?
In On the variation of the spectrum of a normal matrix, Sun proves the following result (Corollary 1.2):
Let $A$ be an $n$-square normal matrix and $B$ an arbitrary $n$-square matrix. Then $$ \min_{\...
- 283
3
votes
0
answers
197
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 ...
- 1,960
1
vote
2
answers
1k
views
Lipschitz continuity of eigenvalues and eigenvectors of Hermitian matrices
It is well-known that the eigenvalues (in decreasing order) of a Hermitian matrix $A$ are Lipschitz continuous functions of $A$.
Do there exist orthonormal eigenvectors that vary in a Lipschitz ...
- 3,433
3
votes
2
answers
368
views
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 ...
- 161
0
votes
1
answer
187
views
Reference for action-angle coordinates [closed]
Does anyone know a good reference to start studying Action-Angle coordinates?
Thank you in advance !
- 97
0
votes
1
answer
160
views
Perturbative approach starting from a probability distribution approximated form
I approximate a probability distribution $P_x(x)$ with a $P_x^{app}(x)$,
such that $P_x(x)-P_x^{app}(x) = O(\epsilon)$ uniformly in x, where epsilon is a small positive quantity.
Consider the generic ...
- 321
2
votes
0
answers
60
views
Stability in coefficients for the rescaled radiative tranport equation
One form of the radiative transport equation is as follows:
$$ v\cdot \nabla_x u + \left(\epsilon \sigma_a(x) + \frac{1}{\epsilon}\sigma_s(x)\right) u - \frac{1}{\epsilon}\sigma_a(x)\int_{S^{n-1}} p(v,...
- 123
1
vote
1
answer
145
views
Sum of positive self-adjoint operator and an imaginary "potential": literature request
To keep things simple, let us consider the following: $L$ is a positive, unbounded S.A. operator on $L_2(\mathbb{R},f(x))$, where $f(x)$ is a Gaussian. Assume that we know the spectrum and ...
- 13
0
votes
0
answers
107
views
Is a basis that almost diagonalizes a matrix 'close' to its eigenbasis?
Let $A\in\mathbb{R}^{n\times n}$ be diagonalisable (over $\mathbb{C}$) with pairwise distinct eigenvalues $\lambda_1,\ldots,\lambda_n\in\mathbb{C}$, and suppose that
$$\tag{1}S^{-1}\cdot A\cdot S = \...
- 453
24
votes
1
answer
5k
views
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 ...
- 413
1
vote
0
answers
154
views
Perturbation in the equation $u_t=\epsilon Pu$, where $P$ is an elliptic partial differential operator
Let $Pu=\sum_{ij} \partial_j(a_{ij}(x) \partial_{i} u)$ be an elliptic operator. Consider the equation
$$
(u=u_\epsilon)\\
\partial_t u=\epsilon Pu \text{ in } \mathbb R^+ \times \Omega,\\
u(0,x)=u_0(...
- 1,795
0
votes
2
answers
359
views
Convergence of the eigenvector matrix for an analytic perturbation of a singular matrix
Let $A$ be an $n\times n$ matrix of all ones. Consider the analytic perturbation of $A$ as $$\tilde{A} = A + \epsilon H_1 + \epsilon^2 H_2 + \epsilon^3 H_3 + ... $$ All matrices are symmetric. Assume $...
- 544
3
votes
0
answers
248
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 (...
- 429
1
vote
1
answer
216
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 ...
- 1,127