Here’s my work:
The integrating factor I find is (x^(2)-1)^1/2. The self adjoint form I find is
-d/dx (((1-x^(2))^(3/2))*dy/dx))=k(x^(2)-1)^(1/2).
Am I right?
Hello,
I hope you are doing well.
I had a question about the eigenvalue problem of quantum mechanics. In a past class, I remember it was strongly emphasized that the eigenvalues of an eigenvalue problem is what we measure in the laboratory.
##A\psi = a\psi##
where A would be the operator...
Can anyone recommend a book in which complex eigenvalue problems are treated? I mean the FEM analysis and the theory behind it. These are eigenvalue problems which include damping. I think that it is used for composite materials and/or airplane engineering (maybe wing fluttering?).
Since the eigenvalue problem can't distinguish between a non-existent wavefunction (and therefore a non-existent particle), and the energy being zero. This is the next thing that has started bothering me on my journey to understand quantum mechanics.
For example, in the algebraic derivation of...
In the symmetric eigenvalue problem, Kv=w^2*v where K~=M−1/2KM−1/2, where K and M are the stiffness and mass matrices respectively. The vectors v are the eigenvectors of the matrix K~ which are calculated as in the example below. How do you find the directions of the eigenvectors? The negatives...
I have a matrix M which in block form is defined as follows:
\begin{pmatrix} A (\equiv I + 3\alpha J) & B (\equiv -\alpha J) \\ I & 0 \end{pmatrix} where J is an n-by-n complex matrix, I is the identity and \alpha \in (0,1] is a parameter. The problem is to determine whether the eigenvalues of...
Good Morning
Could someone give me some numbers for a Generalized EigenValue problem?
I have lots of examples for a 2 x 2, but would like to teach the solution for a 3x3.
I would prefer NOT to turn to a computer to solve for the characteristic equation, but would like an equation where the...
Hi PF!
Here's an ODE (for now let's not worry about the solutions, as A LOT of preceding work went into reducing the PDEs and BCs to this BVP):
$$\lambda^2\phi-0.1 i\lambda\phi''-\phi'''=0$$ which admits analytic eigenvalues
$$\lambda =-2.47433 + 0.17337 I, 2.47433 + 0.17337 I, -10.5087 +...
Hello!
Suppose you have two masses, that are connected by a spring.
Each mass is, in turn, connected by a spring to a wall
So there is a straight line: left wall to first mass, first mass to second mass, second mass to right wall
This problem can be analyzed as an eigenvalue problem.
We...
A theorem from Axler's Linear Algebra Done Right says that if 𝑇 is a linear operator on a complex finite dimensional vector space 𝑉, then there exists a basis 𝐵 for 𝑉 such that the matrix of 𝑇 with respect to the basis 𝐵 is upper triangular.
In the proof, he defines U=range(T-𝜆I) (as we have...
Hi!
I am studying Shankar's "Principles of QM" and the first chapter is all about linear algebra with Dirac's notation and I have reached the section "The Characteristic Equation and the Solution to the Eigenvalue Problem" which says that starting from the eigenvalue problem and equation 1.8.3...
Hi PF!
Given the quadratic eigenvalue problem ##Q(\lambda) \equiv (\lambda^2 M + \lambda D + K)\vec x = \vec 0## where ##K,D,M## are ##n\times n## matrices, ##\vec x## a ##1\times n## vector, the eigenvalues ##\lambda## must solve ##\det Q(\lambda)=0##.
When computing this, I employ a...
Hi PF!
I'm trying to solve the polynomial eigenvalue problem ##M \lambda^2 + \Phi \lambda + K## such that K = [5.92 -.9837;-0.3381 109.94];
I*[14.3 24.04;24.04 40.4];
M = [1 0;0 1];
[f lambda cond] = polyeig(M,Phi,K)
I verify the output of the first eigenvalue via
(M*lambda(1)^2 +...
Homework Statement
An elastic membrane in the x1x2-plane with boundary circle x1^2 + x2^2 = 1 is stretched so that point P(x1,x2) goes over into point Q(y1,y2) such that y = Ax with A = 3/2* [2 1 ; 1 2] find the principal directions and the corresponding factors of extension or contraction of...
Hi PF!
I am trying to solve the eigenvalue problem ##A v = \lambda B v##. I thought I'd solve this by $$A v = \lambda B v \implies\\
B^{-1} A v = \lambda v\implies\\
(B^{-1} A - \lambda I) v = 0 $$
and then using the built in function Eigenvalues and Eigenvectors on the matrix ##B^{-1}A##. But...
Hi PF!
I want to solve ##u''(x) = -\lambda u(x) : u(0)=u(1)=0##. I know solutions are ##u(x) = \sin(\sqrt{\lambda} x):\lambda = (n\pi)^2##. I'm trying to solve via the Ritz method. Here's what I have:
define ##A(u)\equiv d^2_x u## and ##B(u)\equiv u##. Then in operator form we have ##A(u) =...
Hi PF!
Given ##B u = \lambda A u## where ##A,B## are linear operators (matrices) and ##u## a function (vector) to be operated on with eigenvalue ##\lambda##, I read that the solution to this eigenvalue problem is equivalent to finding stationary values of ##(Bu,u)## subject to ##(Au,u)=1##...
The question is posted in the following post in MSE, I'll copy it here:
https://math.stackexchange.com/questions/1407780/a-question-on-matrixs-eigenvalue-problem-from-eberhard-zeidlers-first-volume-o
I have a question from Eberhard Zeidler's book on Non-Linear Functional Analysis, question...
Homework Statement
Using Sturm-Liouville theory, estimate the lowest eigenvalue ##\lambda_0## of...
\frac{d^2y}{dx^2}+\lambda xy = 0
With the boundary conditions, ##y(0)=y(\pi)=0##
And explain why your estimate but be strictly greater than ##\lambda_0##Homework Equations
##\frac{d}{dx} \left...
Homework Statement
Let ##A## be an ##n \times n## matrix. Show that if ##A^2## is the zero matrix, then the only eigenvalue of ##A## is 0.
Homework EquationsThe Attempt at a Solution
All eigenvalues and eigenvectors must satisfy the equation ##A\vec{v} = \lambda \vec{v}##. Multiplying both...
While reading problems in my physics book , I encountered a statement very often "Eigen Value Problem" , I read about it from many sources , but wasn't able to understand it . So what exactly is an Eigen Value Problem?
Homework Statement
find all eigen-values and eigen-functions for the initial boundary value problem:
$$x^2y''+xy'-\lambda y=0$$
Boundary Conditions:
$$y(1)=y(e)=0$$
Homework EquationsThe Attempt at a Solution
i just wanted to know if my substitution in the Auxiliary equation is...
Hi, I'm wondering what eigenvalue problem solver you are using? I'm looking for an one which could solve a very large eigenvalue problem, the matrices being ~ 100,000*100,000. Do you have any advices?
Thanks.
Hi all,
Generally boundary condition (Dirichlet and Neumann) are applied on the Load Vector, in FEM formulation.
The equation i solved, is Generalized eigenvalue equation for Scalar Helmholtz equation in homogeneous wave guide with perfectly conducting wall ( Kψ =λMψ ), and found, doesn't...
Hey! :o
I have the following exercise and I need some help..
$"\text{The eigenvalue problem } Ly=(py')'+qy=λy, a \leq x \leq b \text{ is of the form Sturm-Liouville if it satisfies the boundary conditions } p(a)W(u(a),v^*(a))=p(b)W(u(b),v^*(b)). \text{ Show that the boundary conditions of the...
Homework Statement
Let there be a 4X4 Matrix A with dim(im(A), or rank = 1 , and trace=10. What are the Eigenvalues of A? Are there any multiplicities?
The Attempt at a Solution
While I understand that the trace of a matrix that's 4X4 = the sum of the diagonal elements, I'm confused...
I just have a question about the problem for when the eigenvalue = 0
Homework Statement
for y_{xx}=-\lambda y with BC y(0)=0 , y'(0)=y'(1)
Homework Equations
The Attempt at a Solution
y for lamda = 0 is ax+b
so from BC:
y(0)=b=0
and a=a
What is the conclusion to...
Are eigenvalue problems and boundary value problems (ODEs) the same thing?
What are the differences, if any?
It seems to me that every boundary value problem is an eigenvalue problem... Is this not the case?
I've found the characteristic equation of the system I'm trying to solve:
$$ω^{4}m_{1}m_{2}-k(m_{1}+2m_{2})ω^{2}+k^{2}=0$$
I now need to find the eigenfrequencies, i.e. the two positive roots of this equation, and then find the corresponding eigenvectors. I've been OK with other examples...
Homework Statement
The angular part of the Schrodinger equation for a positron in the field of an electric dipole moment {\bf d}=d{\bf \hat{k}} is, in spherical polar coordinates (r,\vartheta,\varphi),
\frac{1}{\sin\vartheta}\frac{\partial}{\partial\vartheta} \left( \sin\vartheta\frac{\partial...
Hi all,
I want to ask a question about the eigenvalue problem (EVP) and the initial value problem (IVP).
Let's say we are solving this linear equation \frac{\partial u}{\partial t}=\mathcal{L}u, the operator L is dependent on some parameters like Reynolds number.
I first check the...
Hello, I have a feeling that the solution to this question is going to be incredibly obvious, so my apologies if this turns out to be really dumb. How do I solve the following eigenvalue problem:
\begin{bmatrix}
\partial_x^2 + \mu + u(x) & u(x)^2 \\
\bar{u(x)}^2 & \partial_x^2 + \mu +...
Eigenvalue problem after galerkin
Homework Statement
i am working on vibration of cylindrical shell analysis, after solving the equations of motion by galerkin method , i reach to an eigenvalue problem this way :
{ p^2*C1+p*C2+C3 } * X=0
C1,C2,C3 are all square matrices of order n*n...
I know that eigenvalue problem like Lq=\lambda q could be easily solved by eig command in Matlab.
But how to solve a problem like Lq=\lambda q + a, where a has the same dimension with the eigenfunction q?
Thanks a lot in advance.
Jo
Homework Statement
We are to show that for 0<β<1, eigenvalues are strictly positive and for β>1, we have to determine how many negative eigenvalues there are.
u''+λ2u=0, u(0)=0, βu(π)-u'(π) = 0
Homework Equations
I've already shown that the eigenvalues are determined by tan(λπ)=λ/β (was told...
Homework Statement
Hey, guys. I'm having trouble finding the general solution to a second order, homogeneous ODE. It is the first step to solving an eigenvalue problem and my professor is about as much help as a hole in the head. I've tried multiple "guesses" and have combed various...
Homework Statement
Let there be 3 vectors that span a space: { |a>, |b>, |c> } and let n be a complex number.
If the operator A has the properties:
A|a> = n|b>
A|b> = 3|a>
A|c> = (4i+7)|c>
What is A in terms of a square matrix?
Homework Equations
det(A-Iλ)=0
The Attempt...
Consider a generalized Eigenvalue problem Av = \lambda Bv
where $A$ and $B$ are square matrices of the same dimension. It is known that $A$ is positive semidefinite, and that $B$ is diagonal with positive entries.
It is clear that the generalized eigenvalues will be nonnegative. What else can...
Hi all,
What is the normal procedure to verify that I got the correct results (eigenvalues and eigen vectors) from the eigenvalue problem?
I'm using the lapack library to solve eigenvalue problem summarized below. I've 2 matrices K and M and I get the negative results for eigenvalues...
Homework Statement
I have a problem
u'' + lambda u = 0
with BCs: u'(0) = b*u'(pi), u(0) = u(pi).
where b is a constant.
I have to find b which makes the BCs and problem self-adjoint.
Homework Equations
see below
The Attempt at a Solution
I see in my notes...
Hi all,
I need to find the λ and the ai that solves the Generalized eigenvalue problem
[A]{a}=-λ2 [B]{a}
with
[A]=
-1289.57,1204.12,92.5424,-7.09489,-25037.4,32022.5,-10004.3,3019.17
1157.46,-1077.94,-0.580522,-78.9482,32022.5,-57353.5,36280.6,-10949.6...
Homework Statement
Show that the matrix
A = [cos θ -sin θ
sin θ cos θ]
will have complex eigenvalues if θ is not a multiple of π. Give a geometric interpretation of this result.
Homework Equations
Ax = λx, so
det(A-λI) = 0
The Attempt at a Solution
In this case...
Homework Statement
The problem amounts to finding the eigenvalues of the matrix
|0 1 0|
|0 0 1|
|1 0 0|
(I have no idea how to set up a matrix in the latex format, if anyone can tell me that'd be great)
Homework Equations
The characteristic equation for this matrix is...
I have a generalised eigenvalue problem of the form
A\boldsymbol{u} = \lambda B\boldsymbol{u}\;,
where A and B are symmetric matrices with real symbolic entries. I'm trying to compute the eigenvalues with Mathematica using the command
Eigenvalues[{A,B}]
which according to the documentation...
I'm trying to do something that requires solving an eigenvalue problem of the form
A_{imkl} c_m c_k c^*_l=\lambda c_i
where A is a known rank-4 tensor, \lambda is the eigenvalue, and the c_i's are a set of unknown coefficients that I need to determine. I would guess that this type of problem...