# Questions tagged [perturbation]

The perturbation tag has no usage guidance.

34
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### Eigenvalues in matrix perturbation theory converge to wrong value?

Given a Hermitian matrix $H$ and let $\lambda_0^H$ is the minimum eigenvalue of $H$. To use perturbation theory to find $\lambda_0^H$, we can first start with splitting $H$ into something like the ...

0
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1
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119
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### 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-...

2
votes

1
answer

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### 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 ...

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1
answer

115
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### The Eigenvalue Problem: Perturbation Theory

Let $\mathbf{K}$ be a square matrix and $\rho(\mathbf{K})$
is the spectral radius of $\mathbf{K}$. Then, If $\mathbf{M}= \mathbf{K}+\delta \mathbf{A}$ for very small $\delta$, I want to prove that
$$ ...

5
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1
answer

597
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### Perturbation bound for SVD (denoising for a low-rank matrix)

Suppose that $A$ is a $m\times n$ matrix with rank $r$, and we observe the matrix $\hat A = A + E$. Let $\hat A_r$ be the $r$-SVD of $\hat A$. That is, if $A=U\Sigma V^\top$ is the singular value ...

2
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0
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53
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### Two-variable singular perturbation analysis

I am having difficulty with a two-variable singular perturbation analysis on a set of ODE's. The key difficulty is also present in the following, embarrassingly simple problem:
If $x\sim \mathcal{O}(...

3
votes

2
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139
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### Perturbed behavior of a differential equation

Let $a$, $b$ be two real positive parameters with $a>b$, and consider the following nonlinear differential equation:
\begin{align}
\dot{x}_{\varepsilon}(t) = a - b\sin(x_{\varepsilon}(t))+\...

4
votes

0
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78
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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}^{...

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2
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364
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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 ...

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### Perturbation of linear system of equations: Is the solution still non-negative?

Let $A = (a_{ij})_{i,j=1,\dots,n}$ be a matrix such that
$a_{ij} \ge 0$ for all $i,j = 1, \dots, n,$ and
$A$ is positive definite.
Let $I$ be the identity matrix, and $\pmb{1}$ the vector ...

1
vote

0
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173
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### Coefficient perturbations of polynomials with real roots only

Let
\begin{align}
P(x) &= x^n+a_{n-1} x^{n-1} +\ldots+a_0 = \prod_{i=1}^n (x-p_i)\\
Q(x) &= x^n + b_{n-1} x^{n-1} + \ldots +b_0 = \prod_{i=1}^n (x-q_i)\\
p_i, q_i& \in \mathbb{R},\ 0<...

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2
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218
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### Robust generalization of matrix rank

I am looking for robust generalizations of matrix rank.
Think of the the following problem: A big matrix of low rank is perturbed by random noise, such that it becomes a full-rank matrix. Is there a ...

5
votes

2
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298
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### Are periodic billiard trajectories stable on a manifold with strictly convex boundary?

Let $(M,g)$ be a compact Riemannian manifold with strictly convex boundary.
Let $\gamma:S^1\to M$ be a periodic billiard trajectory (geodesic in the interior and reflects specularly at the boundary).
...

6
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298
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### What is the technical difference between a deformation and a perturbation?

What is the technical difference between a deformation and a perturbation? Do they exist in somewhat different categories?

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287
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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 ...

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1
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405
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### How much can I perturb a symmetric stochastic matrix and keep positive solutions?

Suppose $A$ is a symmetric stochastic $n \times n$ matrix with least eigenvalue $\lambda_n>0$. Consider real symmetric perturbations $\widehat{A}$.
How large can I take $\epsilon$ such that $\...

1
vote

2
answers

344
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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 ...

5
votes

1
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321
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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
...

4
votes

1
answer

331
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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 ...

2
votes

0
answers

821
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### Error bound on matrix vector multiplication

I am multiplying a matrix $A$ with vector $p$. However, the matrix $A$ isn't accurate.
Some (a very small fraction) of the element's value is changed from $a_{i,j}$ to {0,$-a_{i,j}$, $2a_{i,j}$}. ...

2
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0
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120
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### Perturbation analysis for three term recurrences

Jacobi polynomials, denoted by $J^{(\alpha,\beta)}_n$, on $[-1,1]$ satisfy a three term recurrence
$$ J_{n+1}^{(\alpha,\beta)}(x) = (A_n+B_nx)J^{(\alpha,\beta)}_n + C_nJ_{n-1}^{(\alpha,\beta)}(x), \...

6
votes

2
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### Eigenvalues of a Symmetric Positive Semi-Definite (PSD) matrix after rank one update

I have a Symmetric Positive Semi-Definite matrix $A$ which i know its eigenvalue and eigenvectors. let $v$ and $u$ be a random column vector. i want to know if it is possible to have eigenvalues of ...

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202
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### Non-linear Perturbation Operator Examples

Consider a non-linear operator $\cal H$ which maps a function to a function (e.g., a map from a starting wave function $f(x,y,z)$ to a later wave function according to some non-linear PDE) and an $\...

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0
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209
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### componentwise eigenvector perturbation

Does the sin-theta theorem imply a componentwise nonasymptotic bound for eigenvectors? Assume, for the purpose of this question, that the eigenvalues concerned are simple.
If this is trivial, I ...

2
votes

1
answer

732
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### Null Space Perturbations

Hi,
I face a problem some time now (not a homework problem) and I believe it is related to matrix perturbations and how the null space behaves in these cases.
The distilled version of the ...

6
votes

2
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656
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### analytic approximation of a non-negative matrix by a sequence of positive matrices

Let $L \in \{0,1\}^{n \times n}$ be a non-negative matrix whose row sum is 1. ($L$ stands for "limit"). It is known that there exists a unique $r \times r$ principal minor of $L$ that is a permutation ...

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201
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### Regular Perturbation Series soln to eqn

I want to find the a 3 term perturbation soln of
(i) $(1+x)^3 = ex$ where $e\ll1$
Direct substitution of the regular perturbation series $x = x_0 + ex_1 + e^2x_2$
into (i) does not work
I ...

3
votes

2
answers

504
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### Number of perturbations of the Jordan form

I am looking for information about the number of Jordan forms that can be obtained from a given Jordan form of a small perturbation.
For example, if a Jordan form consists of a single cell 2x2
$$J=\...

4
votes

1
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425
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### What is known rigoruously about the semiclassical (k to infinity) limit of WRT invariants?

To each closed $3$-manifold $N$, there is a corresponding Witten--Reshetikhin--Turaev invariant $Z_k(N)$ depending on an integer $k$ (the level) and a Lie group $G$ (and perhaps we'll just concentrate ...

3
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1
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324
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### enlarge the separation between two matrices

The separation between two square matrices $A$ and $B$, often used as a measure of the sensitivity of invariant subspace problems, is defined as
$$
\operatorname{sep}(A,B)=\min_{X\neq 0}\frac{\left\...

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3
answers

13k
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### Kind of submultiplicativity of the Frobenius norm: $\|AB\|_F \leq \|A\|_2\|B\|_F$?

Let $\|\cdot\|_F$ and $\|\cdot\|_2$ be the Frobenius norm and the spectral norm, respectively.
I'm reading Ji-Guang Sun's paper 'Perturbation Bounds for the Cholesky and QR Factorizations' from BIT ...

3
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0
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### A question about the generalized Lidskii-Wielandt inequality for matrices proved by Thompson and Freede

In 1971, Thomson and Freede generalized the Lidskii-Wielandt inequalites as follows (version for singular values)
Let $A$, $B$ be $n\times n$ Hermitian matrices. Suppose $\alpha_1\geq \alpha_2 \geq \...

7
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1
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311
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### Adding a multiple of the Identity to a LU factorized matrix

Suppose a square, dense, symmetric matrix $A$ has been factorized into $L$ and $U$ components by performing a LU decomposition. Now let $B = A+\lambda I$. Is there any way to efficiently compute the ...

2
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1
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257
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### Where are the boundary layers?

I am learning perturbation theory and would like to be able to determine where boundary layers are going to occur just by looking at the differential equation.
Let $n\in\mathbb{N}$ and $p_i(x)$, $0\...