Questions tagged [perturbation-theory]
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119 questions
6
votes
2
answers
444
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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
$$ A = \begin{bmatrix}
X & D \\
D^{\top} ...
- 61
0
votes
0
answers
60
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analyzing the sensitivity of two matrix expressions
I'm working on analyzing the sensitivity of two matrix expressions. I'd like to formally show that one is more sensitive to perturbations in a covariance matrix C than the other.
We are given:
$$\...
- 11
5
votes
1
answer
245
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
82
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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
111
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
190
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
152
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
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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
178
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
271
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
159
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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
174
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