# Questions tagged [perturbation]

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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 ...
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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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### 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 ...
264 views

### 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). ...
276 views

### 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?
276 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 ...
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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 $\... 2answers 252 views ### Is Rellich's function valued theorem valid for a rank defficient function valued matrix? Theorem (Rellich). Let$\boldsymbol{A}(t) : \mathbb{R}\rightarrow\mathbb{C}^{n \times n}$be a Hermitian matrix function that depends on$t$analytically. (i) The$n$roots of the characteristic ... 1answer 282 views ### Separating the spectrum of a Hermitian matrix Given Hermitian matrix$A$, I would like to perturbate it so that its eigenvalues become well-separated. Specifically, let$A$be some Hermitian matrix, and let$G$be a Gaussian matrix, with each ... 1answer 232 views ### Distribution of the spectrum of a perturbed matrix Let$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$be a ... 0answers 575 views ### Error bound on matrix vector multiplication I am multiplying a matrix$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}$,$2a_{i,j}$}. ... 0answers 116 views ### Perturbation analysis for three term recurrences Jacobi polynomials, denoted by$J^{(\alpha,\beta)}_n$, on$[-1,1]$satisfy a three term recurrence $$J_{n+1}^{(\alpha,\beta)}(x) = (A_n+B_nx)J^{(\alpha,\beta)}_n + C_nJ_{n-1}^{(\alpha,\beta)}(x), \... 2answers 2k views ### Eigenvalues of a Symmetric Positive Semi-Definite (PSD) matrix after rank one update I have a Symmetric Positive Semi-Definite matrix A which i know its eigenvalue and eigenvectors. let v and u be a random column vector. i want to know if it is possible to have eigenvalues of ... 0answers 201 views ### Non-linear Perturbation Operator Examples Consider a non-linear operator \cal H which maps a function to a function (e.g., a map from a starting wave function f(x,y,z) to a later wave function according to some non-linear PDE) and an \... 0answers 205 views ### componentwise eigenvector perturbation Does the sin-theta theorem imply a componentwise nonasymptotic bound for eigenvectors? Assume, for the purpose of this question, that the eigenvalues concerned are simple. If this is trivial, I ... 1answer 582 views ### Null Space Perturbations Hi, I face a problem some time now (not a homework problem) and I believe it is related to matrix perturbations and how the null space behaves in these cases. The distilled version of the ... 2answers 646 views ### analytic approximation of a non-negative matrix by a sequence of positive matrices Let L \in \{0,1\}^{n \times n} be a non-negative matrix whose row sum is 1. (L stands for "limit"). It is known that there exists a unique r \times r principal minor of L that is a permutation ... 1answer 190 views ### Regular Perturbation Series soln to eqn I want to find the a 3 term perturbation soln of (i) (1+x)^3 = ex where e\ll1 Direct substitution of the regular perturbation series x = x_0 + ex_1 + e^2x_2 into (i) does not work I ... 2answers 475 views ### Number of perturbations of the Jordan form I am looking for information about the number of Jordan forms that can be obtained from a given Jordan form of a small perturbation. For example, if a Jordan form consists of a single cell 2x2$$J=\... 1answer 405 views ### 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 ... 1answer 308 views ### 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\... 3answers 9k views ### 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 ... 0answers 413 views ### 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 \...
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 ...
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\...