2
votes
0answers
78 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 ...
1
vote
2answers
58 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. ...
4
votes
1answer
36 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 ...
0
votes
0answers
16 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 ...
0
votes
0answers
30 views
+50

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 ...
0
votes
0answers
24 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 ...
6
votes
2answers
472 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
votes
0answers
23 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: $-X''=\lambda X$ $X'(0)-a_0X(0)=0$ ...
0
votes
0answers
15 views

Eigenvalues and fft of sound trying to find similarities

I'm looking at sound resonance patterns using matlab/octave to see if there may be patterns between FFT and Eigenvalues. I can get the frequencies and each of the frequencies amplitude to recreate ...
0
votes
0answers
23 views

Removing extraneous solutions from an eigenvalue equation

I have an eigenvalue problem of the form $\left[ L_1 + \dfrac{L_2}{\Omega} + \dfrac{L_3}{\Omega^2} + \dfrac{\Omega-1}{\Omega+\eta}\right] \phi(x) = 0$ which I am trying to solve for the complex ...
1
vote
1answer
24 views

Finding linear transform matrix from characteristic polynomial

I got two similiar very simple question on a notebook. 1)let characteristic polynomial $P_A(x)=x^2+2x-3$ and $T:V\to V$ and DimV=2,S={$\alpha_1,\alpha_2$} is ...
2
votes
1answer
20 views

Estimation with an orthonormalbasis in some finite dimensional subspace of $L_2(\Omega)$

I'm currently trying to understand a step of a proof of the following estimation: $\displaystyle \sum_{j=2}^{N} \left(\left< \varphi_1-R_N\varphi_1, \varphi_{j,N} \right>_{L_2}\right)^2 \ ...
0
votes
1answer
46 views

Proving subspaces are invariant for different statements [closed]

I would like to know how to prove the next statements regarding invariant subspaces: Statement 1: $f$ and $g$ are endomorphisms from a vector space $V$. If $f$ and $g$ commute, then subspaces ...
2
votes
0answers
28 views

Eigenfunctions of a 2D fractional Brownian motion covariance

The fractional Brownian motion is a centered Gaussian process with the following covariance function (covariogram): $E[B(t)B(s)]=C(\Vert t \Vert ^{2H}+\Vert s\Vert^{2H}-\Vert t-s\Vert^{2H})$ ...
0
votes
1answer
41 views

General eigenspace & general eigenvector

Consider the following: 1.Suppose $k_i$ is an eigenvalue of $A$ with algebraic multiplicity $n_i$ 2.dim $V_i = m_i$ (geometric multiplicity), $V_i$ is the eigenspace corresponding to $k_i$. 3. $m_i ...
2
votes
1answer
34 views

Eigenvalue of (1-0) matrix

Assume I have 2 matrices, each of size nxn with only 1 and 0 as entries in both. (n>10) The first matrix (call it A) has each row summing up to 2 (ie: on each row, it has two "1" and n-2 "0"). It is ...
0
votes
2answers
36 views

Eigenvalue of a linear transformation

Let $V$ be the linear space of all real polynomial $p(x)$ of degree $\leq n$.If $p \epsilon V$, define $q=T(p)$ to mean that $q(t)=p(t+1)$ for all real $t$. Prove that $T$ has only the eigenvalue $1$. ...
2
votes
1answer
46 views

the solution of matrix polynomials

In order to get the eigenvalues of \begin{equation} P=\left[ \begin{array}{cc} 0_{n\times n} & I_{n\times n} \\ -A & -B% \end{array} \right], \end{equation} where $A$ and $B$ are both $n\times ...
2
votes
4answers
83 views

Stuck in finding Eigen values

The given matrix A is $$ \left[\begin{matrix} 2 & 1 & -2 \\ 0 & 1 & 4 \\ 0 & 0 & 3 \\ \end{matrix}\right] $$ I know that the Eigen values are the diagonals (2, 1, 3) as it is ...
2
votes
0answers
187 views

Eigenvalues of self-adjoint eigenvalue problem

I am stack with the following problem: Consider the following eigenvalue problem $$ u \in H_B(0,1), \; \langle Lu, Lv\rangle = \lambda (\alpha \langle u, v\rangle + \langle u', v'\rangle) \; \forall ...
0
votes
1answer
57 views

Completeness of eigenvectors of Hermitian Matrix.

How do you show that eigenvectors of a Hermitian matrix form a complete set of basis?
0
votes
0answers
38 views

Convert an eigenvalue equation to ODE/s

For example define: $K=-i\frac{d}{dx}$ (non-discrete spectrum), so: $$Kf(x)=-i\frac{df}{dx}=kf(x)$$ Define $g(x,k)=kf(x)$, so: $$\frac{-i}{k}\frac{\partial{g}}{\partial{x}}=g(x,k)$$ ...
0
votes
0answers
52 views

Regular Sturm-Liouville Boundary Value Problem

Let $L[y]:=y''''$. Let the domain of $L$ be the set of functions that have four continuous derivatives on $[0,π]$ and satisfy $y(0)=y'(0)=0$ and $y(π)=y'(π)=0$ a) Show that $L$ is self adjoint b) ...
1
vote
1answer
166 views

On max-min representation for the principal eigenvalue of second order elliptic operator

(Just to be upfront about things, this is a homework problem.) I'm asked to show that the principal eigenvalue, $\lambda_1$ of an uniformly elliptic operator can be represented by \begin{equation} ...
1
vote
1answer
102 views

Eigenfunction expansion

Use the appropriate engenfunction expansion to represent the best solution. $$u''=f(x), u'(0)=\alpha, u'(1)=\beta$$ I use the function $$\phi''+\lambda\phi=0$$ to get the eigenfunction is ...
2
votes
0answers
80 views

What are the real world uses of Eigenbasis

The title pretty much says it all, I am wondering what the real world application (especially pertaining to electrical engineering) of an Eigenbasis is. I am also having some trouble understanding ...
4
votes
1answer
150 views

What can be said about the eigenvalues of the Laplace operator in $H^k(\mathbb{T}^2)$

Consider the Laplace operator $$\Delta: H^{k+2}(\mathbb{T}^2) \to H^k(\mathbb{T}^2)$$ where $\mathbb{T^2}$ is the two-dimensional torus (which is a compact manifold without boundary), so that $$ ...
2
votes
0answers
49 views

Given two eigenfunctions and eigenvalues determine existence of eigenvalues between them

Suppose we have two eigenfunctions $f_n(x,y)$ and $f_m(x,y)$ and corresponding eigenvalues $\lambda_n<\lambda_m$ of a differential operator $L$. How can I determine whether there exists another ...
2
votes
1answer
188 views

Arbitrarily using Sin and Cos as eigenfunctions of a Hamiltonian?

In the context of quantum optics, the rotating wave Hamiltonian can be written: $\hbar\begin{pmatrix} -\Delta & \Omega/2\\ \Omega/2 & 0 \end{pmatrix}$ The eigenvalues can then be calculated ...
0
votes
1answer
115 views

Eigenvalues of a second derivative

I have a function f(r) that describes a Gaussian random field. A second derivative can be formed $\nabla_i \nabla_j f(r)$. I am looking at a paper that claims that in finding the extremum, the ...
0
votes
1answer
406 views

Finding eigenvalues and eigenfunctions for a BVP

Find the eigenvalues and eigenfunctions for $$y'' + \lambda y = 0, y(0) = 0, y'(\pi/2) = 0$$ According to my book we must check 3 cases: $\lambda < 0$, $\lambda = 0$, $\lambda > 0$. I started ...
0
votes
2answers
240 views

Problem related with boundary value problem and eigenvalue, eigenfunctions

I was looking at previous year exam papers and was stuck on the following problem: For the boundary value problem, $\,\,y''+\lambda y=0; y(0)=0,y(1)=0, \,\,\exists$ an eigenvalue $\lambda$ ...
1
vote
1answer
74 views

Show that eigenvalues are negative

I have to consider the eigenvalue problem: $$ L[u] := \frac{d^2 u}{dx^2}= λu,x \in (0,1)\quad u(0)-\frac{du}{dx}(0)=0, u(1)=0.$$ I need to show that the eigenvalues are negative.
1
vote
2answers
78 views

Upper bound on the difference between two elements of an eigenvector

Let $W$ be the non-negative, symmetric adjacency/affinity matrix for some connected graph. If $W_{ij}$ is large, then vertex $i$ and vertex $j$ have a heavily weighted edge between them. If $W_{ij} = ...
1
vote
0answers
101 views

Completeness of eigenfunctions

In my computations I have obtained a sequence of eigenvalues $\lambda_k, \; k\in \mathbb{N}$ of double multiplicity. Thus, the basis for the eigenspace of $\lambda_k$ is given by $\psi_k(x) = ...
1
vote
0answers
38 views

First eigenvalue of the given linear operator

I have the following question: Let us denote $H_2^N: = \{u\in (H^2(0,1))^2: u'(0) = u'(1) = 0\}$. Let an operator $L:H_2^N \to (L^2(0,1))^2$ be given by $Lu = -Du'' + Cu$, where $D$ is a positive ...
3
votes
1answer
225 views

Eigenvectors and Principal component

What is the difference between eigenvectors and principal component. I got confused about this point because some researches reported that the principal components are the same eigenvectors of ...
1
vote
1answer
221 views

How do you find the (complex) eigenvalue and each eigenspace over C

My book only has eigenvalue and eigenspace and does not say anything about complex eigenvalue and eigenspace. $$A = \begin{pmatrix} 0&4\\-1 & 0 \end{pmatrix}\hspace{10pt}B =\begin{pmatrix} ...
1
vote
0answers
90 views

The value interpretation of eigenvectors.

My question is may be strange but I wanna lie it any way. The direction of an eigenvector is the most important as we normalize it. This view is right but what about the value of this eigenvector in ...
2
votes
3answers
226 views

Spectrum of eigenvalues and eigenfunctions

Our O.D.Es professor had the "amazing" idea of heavily introducing advanced linear algebra material (which is not an official prerequisite for the course) along with boundary value problems. Not being ...
0
votes
1answer
165 views

Eigenvalues and Eigenvectors, difference between integer results and absolute results

i am developing an application where i need to calculate the eigenvalues and it's corresponding eigenvectors,i understand how to calculate it from this links: ...
1
vote
1answer
98 views

Which one is the right definition of eigenvalues for a differential operator?

This question might be trivial, but I have problems understanding the definition of eigenvalues for the Laplacian \begin{equation} \Delta : C^2(U) \to C(U). \end{equation} on some open, bounded domain ...
3
votes
0answers
117 views

show that the function satisfies condition of the lemma

Let $(f_n)_{n\geq 0}$ be an eigenfunctions of the compact integral operator $F$, defined on $L^2([-1,1])$ by $$ F(f)(x)=\int_{-1}^1\frac{\sin a(x-y)}{x-y}\,\psi(y)\,\mathrm{d}y, \quad \mbox{here ...
1
vote
1answer
326 views

degenerate eigenvalues

I have a problem in understanding the exact meaning of degenerate eigenvalue. I have some database and I calculate the covariance matrix among it. the obtained eigenvalues are same ( all of them ...
3
votes
1answer
644 views

Rayleigh-Ritz Theorem

Let $U$ be an $n$-dimensional subspace of $L:=L_2([-1,1])$. Let $F$ be an acting on $L$, given at $f \in L$ $$ (Ff)(x):=\int_{-1}^1 \frac{\sin a(x-y)}{(x-y)}f(y) dy, \quad x \in [-1,1], \quad a>0. ...
1
vote
2answers
686 views

Show that the function is an eigenfunction of the equation

I'm not sure how to use the bbcode so I've taken a screenshot instead: Came up on a past exam paper that I'm working towards and I'm not sure how to answer it. I assumed that EQN . EIGENFUNCTION ...
4
votes
1answer
510 views

How to find an orthonormal basis for $L^2(\mathbb{R},\mathbb{C})$?

Consider the Hilbert space $X:=L^2(\mathbb{R},\mathbb{C})$ Now consider the operator that takes the second derivative, i.e. $A := \partial_{x}^2$, i.e. $A: H^2(\mathbb{R},\mathbb{C}) ...
1
vote
0answers
90 views

Partial Differential Equation Eigenvalue of zero question

In the event that I'm solving a partial differential equation through separation of variables, if I end up with an eigenvalue of zero, what do I do with the corresponding eigenfunction? That is to ...
2
votes
1answer
1k views

Difference between eigenfunctions and eigenvectors of an operator?

What is the difference between the eigenfunctions and eigenvectors of an operator, for example Laplace-Beltrami operator?
1
vote
1answer
114 views

Eigenstate and quantum mechanics position opperator

Quantum mechanics math question: Suppose that there is eigenstate $|q \rangle$ where $q$ is position observable . The question is, 1) What is eigenstate? How is this different from eigenvector? ...