The perturbation-theory tag has no usage guidance.

**2**

votes

**0**answers

29 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 ...

**0**

votes

**0**answers

22 views

### Weighted Perturbation Bound for Polar Decomposition

Setup: Let $X\in\mathbb{R}^{n\times r}$ be a matrix with orthogonal columns, with $\Sigma = X^TX$, and assume that $\Sigma$ is invertible (note, $\Sigma$ is not necessarily the identity).
Suppose we ...

**1**

vote

**0**answers

68 views

### 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, ...

**0**

votes

**0**answers

46 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 ...

**0**

votes

**1**answer

81 views

### 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
...

**1**

vote

**1**answer

94 views

### 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 ...

**0**

votes

**1**answer

49 views

### 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 ...

**1**

vote

**1**answer

203 views

### 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

82 views

### 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

80 views

### 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 ...

**2**

votes

**0**answers

207 views

### 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 ...

**0**

votes

**2**answers

126 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 ...

**5**

votes

**1**answer

218 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

564 views

### 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

126 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 ...

**2**

votes

**0**answers

71 views

### series representation for *un*bounded 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 ...

**5**

votes

**0**answers

148 views

### 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

84 views

### 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 ...

**1**

vote

**1**answer

149 views

### Weyl asymptotics vs. form perturbations

Consider Hilbert spaces $V$,$H$; a closed quadratic form $a$ with domain $V$; and its associated operator $A$ on $H$. (If necessary, the form can be assumed to be coercive.) For the sake of ...

**5**

votes

**0**answers

293 views

### Rigorous results on the method of multiple scales

The method of multiple scales (Scholarpedia) is a technique used to obtain approximate solutions to differential equations, most commonly when some of the more standard approaches to perturbation ...

**13**

votes

**2**answers

1k views

### Minimum off-diagonal elements of a matrix with fixed eigenvalues

Hello,
I am en engineer working in radar research. I came accross a problem I cannot seem to find math literature on it.
I can ask it in two different ways. Perhaps depending on the reader, the ...