For questions on eigenfunctions.

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1answer
43 views

Eigenvalues and eigenfunctions of fourth order ODE

Find the eigenvalues and eigenfunctions of the problem $$y^{(4)} − λy = 0$$ with the boundary conditions (i) $\quad y(0) = y'' (0) = y(β) = y'' (β) = 0$ (ii) $\quad y(0) = y' (0) = y'' (β) = y''' ...
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0answers
13 views

Does the magnus convergence test not hold for the factorization of second order differential operators?

Given the operator \begin{align} H = V(x)-\partial_x^2 \end{align} and given an eigenfunction $\phi_0(x)$ such that $H\phi_0=0$ with a zero eigenvalue, I can factor $H$ into \begin{align} H = h_+h_- ...
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0answers
15 views

Eigenelements of Neumann Laplacian satisfy $\sum_{k=1}^\infty |(u,\varphi_k)_{L^2}|^2 \lambda_k^{-\frac 12} < \infty?$

Let the eigenvalues of Neumann Laplacian on a bounded open domain be given by $0 = \lambda_0 \leq \lambda_1 \leq \lambda_2 ...$ associated to eigenfunctions $\varphi_0, \varphi_1, ...$. Let $u \in ...
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1answer
44 views

Space of continuous functions linear operator eigenvalues

Let $V$ be the vector space of continuous functions from $\mathbb R$ to $\mathbb R$. Let $T$ be the linear operator on $V$ defined as $$(Tf)(x)=\int_0^x f(t)\,dt$$ Prove that $T$ doesn't have ...
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0answers
22 views

Questions about eigen vectors calculation in eignfaces

I am calculating the eigen vectors from a set of M grayscale faces images and I am using two methods. I would expect to get the same results but calculation gave me the different outcomes. Suppose I ...
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2answers
24 views

Why does the set of an hermitian operator's eigenfunctions spans the functions space

During a discussion about linear hermitian operators, my professor claimed that if a linear operator $M$ is hermitian under a certian set of conditions, then genrally any function that fulfills these ...
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1answer
23 views

Solving 2D Laplace eigenfunction equation

I want to solve the equation $$\nabla^{2}\,{\rm P}(x, y) = \frac{k}{c^{2}}{\rm P}(x, y)$$ or $$\frac{\partial^{2}{\rm P}}{\partial x^{2}}+\frac{\partial^{2}{\rm P}}{\partial y^{2}}=\frac{k}{c^{2}}{\rm ...
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1answer
30 views

Can Eigen vectors be different with the same normalization proceduce?

I calculated Eigen vectors of two badly-conditioned symmetric matrices of $K$ and $M$ ($M$ is positive definite). I employed two algorithms, the 1st algorithm is ...
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1answer
39 views

Can you help with the Method Of Eigenfunction Expansion of a Non-Homogeneous PDE problem?

Here is the Problem: Solve $\frac{\partial T(x,t)}{\partial t} = \frac{\partial^{2} T(x,t)}{\partial x^{2}} +2xe^{-t} $ with the following boundary conditions $T(0,t)=10, and \frac{\partial ...
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1answer
27 views

Eigenvalue problem $y'' + \lambda y = 0,$ $y'(0) = 0$, $y(1) = 0$

Find the eigenvalues of $$y'' + \lambda y = 0, \; y'(0) = 0, y(1) = 0$$ For $\lambda >0$, $$y(x) = c_1 \cos(\sqrt{\lambda} x) + c_2 \sin(\sqrt{\lambda}x)$$ We get that $y'(0) = 0 \implies ...
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1answer
57 views

How to change property of Eigen vectors

I calculated the Eigen values & vectors of two $K$ and $M$ matrices with LAPACK DGGEVX routine. I need to make Eigen vectors orthonormal like this: $\Phi^T M \Phi = I$ . How can I do that?
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0answers
40 views

Neumann eigenvalue problem for the Laplacian

Let $\Omega$ be a bounded, connected open subset of $\mathbb R^n$. Consider the Neumann eigenvalue problem $$ \Delta u =\lambda u \, \text{at} \, \Omega \\ \frac{\partial u}{\partial \vec{\nu}}=0 ...
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0answers
17 views

Helmholtz equation in 3D

I understand solving the Helmholtz equation, $\nabla^2u + \lambda^2u = 0$, when $\nabla^2 = \partial_{xx} + \partial_{yy}$. However, I am not sure of the form I should obtain when $\nabla^2 = ...
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3answers
50 views

EigenFunction for $\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x} =\frac{2f^2}{x}$

When studying a computer vision problem I end up with a function $f(x,t)$ that satisfying $\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x} =\frac{2f^2}{x}$. My question includes two ...
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2answers
47 views

Finding Fourier cosine series of sine function

I am trying to find Fourier cosine series of following function, but think that I am messing up somewhere. $$ f(x)=\sin \bigg ( \frac{\pi x}{l} \bigg ) $$ Fourier cosine series can be written as $$ ...
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1answer
39 views

Finding eigenvalue and eigenfunction of the boundary value problem

How to find eigenvalues and eigenfunctions of this boundary value problem? $$ y'' + \lambda y = 0 \\ y'(0)=0, y(\pi/2)=0 $$ I want to find only positive eigenvalues. I proceed like this: $$ y=C_1 ...
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1answer
46 views

How to find eigenvalues and eigenfunctions of this boundary value problem?

I want to find eigenvalue and eigenfunction of this problem: $$ y''+ \lambda y=0, 0<x<l \\ y(0)=0, ly'(l)+ky(l)=0 $$ And $y'$ stands for $\frac{dy}{dx}$ and similar for $y''$. I get the ...
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1answer
25 views

How to find solution of this eigenfuction?

The eigen function boundary value problem is ($y'=\frac{dy}{dx}$ and similar for $y''$) $$ y'' - \lambda y = 0,\\ y'(0) = y(2)=0 $$ I think the solution is like this Case 1: $\lambda <0 $ putting ...
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0answers
17 views

Can you help me on this Sturm-Liouville Problem?

Here is the problem: If $L$ is the following first order linear differential operator, $L = p(x) \frac{\partial}{\partial x} $, then determine the adjoint operator, $L^*$ such that $\int_{a}^{b} ...
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0answers
24 views

Imaginary Eigenvectors and Systems

NOTE: This is a homework problem (not graded). I want to learn how to do this math problem, not just be given an answer. Thank you for any help you can give! So, in this problem I am dealing with ...
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0answers
33 views

Expansion in Chebyshev polynomials

Please refer to attached image at bottom of page I did an assignment earlier, and I got this question wrong. The solutions have been put up, but I struggle to understand how to proceed from $1^*$ to ...
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1answer
46 views

Karhunen-Loève Expansion (Finding the Eigenfunctions)

A WSS random process $X(\mu,t)$ has $\eta_X(t) = 0$ and $R_X(\tau) = \sum_{k=1}^K\cos(\frac\pi k\tau)$ for a given positive integer $K$ in the interval $|t|<1$. Find the Karhunen-Loève expansion of ...
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0answers
47 views

Accelerated Eigenfunction Expansions of Random Functions

I am interested in eigenfunction expansions of random functions. We know that the autocorrelation of brownian motion, $\{ B_t \}_{t \geq 0}$, is given by $$ E[B_t B_s] = \min\{s,t \}, $$ which can ...
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0answers
11 views

Projection matrix I_V*V', can it change null space of orignial matrix?

I have a matrix, M. suppose i have null(M)=10 and rank(M)=50. the eigenvectors of M are v1-v10 represents the null space and v11-v50 represent the other eigenvalues. now i have a projection matrix ...
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0answers
25 views

Eigenfunctions of the exponential of the derivative

In the space of smooth functions of one variable is there a way to tell what are the eigenfunctions of the operator $\exp(\partial_x)$, i.e. what is the solution of the eigenvalue problem ...
2
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1answer
59 views

Linear Algebra question relating to eigenvectors

Let A be an m x m positive definite symmetric matrix with eigenvalue-eigenvector pairs $(\lambda_1,e_1),....,(\lambda_m,e_m).$ The eigenvectors are orthonormal. Let $C = e_1e_1'+....+e_me_m'$. ...
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0answers
28 views

Conceptual difficulty about eigenfunction expansions

I'm having a fairly large conceptual block on my understanding of eigenfunctions - I've tried out two different methods that yield embarrassingly different answers, and I would appreciate if someone ...
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1answer
55 views

Poisson's Equation in a Disk

How can we solve Poisson's equation in a disk in plane polar coordinates?: $$ \nabla^2 \phi = u_{rr} + \frac{1}{r}u_r + \frac{1}{r^2} u_{\theta \theta} = f(r, \theta)$$ (My attempt): We know that ...
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0answers
41 views

eigenfunctions for integral operator with difference domains

I have a kernel $k(x,y)>0$ ($x$ and $y$ are positive real numbers.), a given eigenvalue $\lambda>0$ and a constant $A$, I'm looking for the condition for $k(x,y)$ such that the Fredholm ...
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1answer
51 views

Why does an eigenvalue expansion 'work' for PDEs?

I understand the logic and rationale behind using a series of eigenfunctions to represent general solutions to simple partial differential equations with prescribed boundary values, such as the ...
2
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1answer
34 views

eigenfunctions on covering spaces of graphs

I am reading about lifts of graphs in relation to covering spaces. Before I pose my question I will explain some of the terminology. Let $G$ and $H$ be two graphs. We say that a function $f: V(H) ...
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1answer
60 views

Showing that two functions are orthogonal on a rectangle

I was given the following question, and I think I'm nearly there, I just wanted to ask for some clarification in the last step. Derive the eigenvalues and functions of the SL problem $\phi_{xx} ...
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3answers
57 views

Basis of eigenfunctions in Banach space

I have a question about proving the existence of a basis of eigenfunctions. Assume we have a compact operator $L:\mathcal{B}\rightarrow\mathcal{B}$, where $\mathcal{B}$ is a Banach space of analytic ...
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0answers
53 views

Finding eigenfunctions of integral operator, numerical method.

I'm trying to find the eigenfunctions and eigenvalues, $b_l$ and $B_l(\eta)$ of an integral operator. This is mentioned in a recent signal processing paper, "Reduced Mean-Square Error Quadratic ...
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0answers
14 views

What properties must a function space satisfy to be able to express any function form?

Given a function space of $X_{n}(x)$ , what properties must the space satisfy so that any mathematical function of two variables can be represented in the following form? $$ ...
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0answers
18 views

Sturm-Liouville equation with rational coefficient

I am trying to solve a regular Sturm-Liouville type Ordinary Differential Equation (ODE) with the following form: \begin{equation} \frac{d}{dy} [q(y) \frac{dW(y)}{dy}]+p(y) W(y) = -\lambda W(y) ...
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1answer
88 views

How to write a non-homogeneous equation in self-adjoint form

How can I write a non-homogeneous equation in self-dajoint form? such as, for equation with $-1\le x \le1$ $$(1-x^2)u''-xu'+2u=x^4+x$$ What is its self-dajoint form? Also, for a ...
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0answers
90 views

Same eigenvalue spectrum with different matrices

There are two matrices $D$ and $T$ which provides eigenvalue spectrum (dispersion relation) according to $$ E(K) = \operatorname{eig}(D + T \exp(iK) + T' \exp(-iK)) $$ $$ K = 0:dK:\pi $$ Where K is a ...
2
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0answers
89 views

Spectrum of $Tu=\int^1_{-1} (1-|x-y|)u(y)dy $

Consider the operator $$ Tu(x)=\int^1_{-1} (1-|x-y|)u(y)dy $$ We want to find the spectrum of $T$. The kernel is certainly bounded and so this operator is Hilbert-Schmidt, so $T$ is compact. We ...
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2answers
113 views

Eigenvalues of symmetric elliptic operators

As stated in one of my previous questions already, I have not had so much exposure to theoretical linear algebra. This time, I'm reading a theorem and proof from PDE Evans, 2nd edition, pages 335-356. ...
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1answer
66 views

What is the purpose of studying Sturm-Louville eigenvalue problem?

After a cursory read on the SL eigenvalue problem, I did not immediately feel enlightened and failed find much usefulness except for knowing that SL generalizes a broader class of differential ...
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0answers
24 views

Eigenvalues of correlation matrices in the limit of infinite dimensions

Consider a continuous function $f(x,t)$ with $x\in X$ and $t\in[0,1]$, then one may define a series of functions $f_n\in\mathbb{R}^n$ defined naturally as $f_n(x)_i=f(x,i/n)$. Now compare the ...
2
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1answer
97 views

Block diagonalizing two matrices simultaneously

There are two matrices $A$ and $B$ which can not be diagonalized simultaneously. Is it possible to block diagonalize them? What if the matrices have an special pattern? Physics of the problem is ...
2
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2answers
42 views

Spectrum of $Tu=\int^1_0 (x+y)u(y)dy$

Given the operator $$Tu(x)=\int^1_0 (x+y)u(y)dy$$ on $L^2(0,1)$, find the spectrum of $T$. For all eigenvalues, find their multiplicities and the eigenfunctions. The kernel is Hilbert Schmidt ...
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0answers
25 views

Which casses of matrices contain A and which contain B? Linear Algebra

Am pretty confused about classes, I don't know what it means, so so I can't really do part_A and I need your help with it? For part B, I got all eigen = 1 for matrix A, and 0 for matrix B, Is this ...
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1answer
57 views

Fourier-Transformation of Operator

I have an operator $\hat{L}$ which gives $$\hat{L} f(x) = \lambda \cdot f(x)$$ where $\lambda$ is the eigenvalue. Now I Fourier-Transform my function $f(x)$: $$\mathcal{F}(f)(p) = g(p)$$ Question: ...
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2answers
145 views

Eigenvalue and Eigenfunction for a boundary value problem

I'm having trouble understanding some of the concepts related to these problems. Here's an example I'm working on: $$y''+(\lambda+1)y=0 ; y'(0)=0,y'(1)=0$$ The characteristic equation I found was ...
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2answers
483 views

Why integration operator has no eigen values?

Let $V$ be the vector space of all functions from $\mathbb R$ into $\mathbb R$ which are continuous. Let $T$ be the linear operator on $V$ defined by $$(Tf)(x) = \int_0^x f(t) dt$$ Prove that ...
2
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0answers
51 views

eigen value problem with Robin Boundary Conditions at both ends

This is a problem from the book Partial Differential Equations by Walter.A.Strauss. Consider the eigen value problem with Robin Boundary Conditions at both ends: $-X''=\lambda X$ $X'(0)-a_0X(0)=0$ ...
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1answer
45 views

Bound on number of mutually orthonormal eigenfunctions

Let $E$ be the vector space of real valued continuous functions on an interval $[a,b]$. Let $K = K(x,y)$ be a continuous function of two variables, defined on the square $a \leq x \leq b$ and $a \leq ...