Eigenvalue Definition and 402 Threads

Hey! :o Let $G$ be the iteration matrix of an iteration method. So that the iteration method converges is the only condition that the spectral radius id less than $1$, $\rho (G)<1$, no matter what holds for the norms of $G$ ? I mean if it holds that $\|G\|_{\infty}=3$ and $\rho (G)=0.3<1$ or...

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) =...

I did an exercice for an optic course and the question was to find which optical component, using eigenvalues and eigenvectors, the following Jones matrix was (the common phase is not considered) : 1 i i 1 I found that this is a quarter-wave plate oriented at 45 degree from the incident...

Hey! :o We have \begin{equation*}A:=\begin{pmatrix}-5.7 & -61.1 & -32.9 \\ 0.8 & 11.9 & 7.1 \\ -1.1 & -11.8 & -7.2\end{pmatrix} \ \text{ and } \ z^{(0)}:=\begin{pmatrix}1\\ 1 \\ 1\end{pmatrix}\end{equation*} I want to approximate the biggest (in absolute value) eigenvalue of $A$ with the...

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##...

Homework Statement Show that if ##A^2## is the zero matrix, then the only eigenvalue of ##A## is 0. Homework Equations ##Ax=λx##. The Attempt at a Solution For ##A^2## to be the zero matrix it looks like: ##A^2 = AA=A[A_1, A_2, A_3, ...] = [a_{11}a_{11}+a_{12}a_{21}+a_{13}a_{31} + ... = 0...

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 Consider an n x n matrix A with the property that the row sums all equal the same number S. Show that S is an eigenvalue of A. [Hint: Find an eigenvector.] Homework Equations ##Ax=λx## The Attempt at a Solution S is just lambda here, so I begin solving this just like you...

Homework Statement Find the eigenvalues of the matrix ##\begin{bmatrix} 4 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & -3 \end{bmatrix}## Homework Equations ##Ax=λx## The Attempt at a Solution I'm having some trouble finding the eigenvalues of this matrix. The eigenvalue of a matrix is a scalar λ such...

Homework Statement Homework Equations The Attempt at a Solution [/B] ##\hat{S}_1.\hat{S}_2 = (S(S+1) - S_1(S_1 + 1) - S_2(S_2 + 1))/2## therefore singlet: ##\psi_s = \frac{\phi_a (1)\phi_b(2)(\alpha(1)\beta(2) - \alpha(2)\beta(1))}{\sqrt(2)}## So for singlet, ##\mathcal{V} = -\frac{K...

Homework Statement Finding eigenvalues of an hamiltonian Homework EquationsH = a S²z + b Sz (hbar = 1) what are the eigenvalues of H in |S,M> = |1,1>,|1,0>,|1,-1> The Attempt at a SolutionH|1,1> = (a + b) |1,1> H|1,0> = 0 H |1,-1> = (a-b) |1,-1> which gives directly the energy : a+b , 0 ...

There is an operator in a three-state system given by: 2 0 0 A_hat = 0 0 i 0 -i 0 a) Find the eigenvalues and Eigenvectors of the operator b) Find the Matrix elements of A_hat in the basis of the eigenvectors of B_hat c) Find the Matrix Elements of A_hat...

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 Homework Equations The Attempt at a Solution When I take the second formula, multiply by it's conjugate and then by x and do the integral of the first formula, I get 0, and not L/2, for <x>. Am I missing a formula ? The complex conjugate of the exponential part...

Homework Statement Given a matrix M={{2,1},{1,2}} then value of cos( (π*M)/6 )Homework EquationsThe Attempt at a Solution Eigen values are π/6 and π/2 and eigen vectors are (π/6,{-1,1}) and (π/2,{1,1}). Diagonalize matrix is {{π/6,0},{0,π/2}} I got same value (√3/2)M

1. ... Expand the Eigenvalue as a power series in epsilon, up to second order: λ=[3+√(1+4 ε^2)]V0 / 2 Homework Equations I am familiar with power series, but I don't know how to expand this as one.[/B]The Attempt at a Solution :[/B] I have played around with the idea of using known power...

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...

1. 1) Given 2x2 matrix A with A^t = A. How many linearly independent eigenvectors is A? 2) Is a square matrix with zero eigenvalue invertible? 2; When it comes to whether it is invertible; the det(A-λ* I) v = 0 where det (A-λ * I) v = 0 where λ = 0 We get Av = 0, where the eigenvector is...

Homework Statement I'm currently working on a homework set for my intermediate QM class and for some reason I keep drawing a blank as to what to do on the first problem. I'm given three potentials, V(x), the first is of the form {A+Bexp(-Cx^2)}, the others I'll leave out. I'm asked to draw the...

I am facing some difficulties solving one of the questions we had in our previous exam. I am sorry for the bad translation , I hope this is clear. In each section, find all approppriate matrices 2x2 (if exists) , which implementing the given conditions: is an eigenvector of A with eigenvalue...

Homework Statement Consider the following matrix A (whose 2nd and 3rd rows are not given), and vector x. A = 4 4 2 * * * * * * x = 2 -1 10 Given that x is an eigenvector of the matrix A, what is the corresponding eigenvalue? Homework EquationsThe Attempt at a Solution 4−λ 4 2 a...

Hey guys, I got the following derivation for some physical stuff (the derivation itself is just math) http://thesis.library.caltech.edu/5215/12/12appendixD.pdf I understand everything until D.8. After D.7 they get the eigenvalue and eigenvectors from ε. The text says that my δx(t) gets aligned...

Hello. If I represent a vector space using matrices, for example if a 3x1 vector, V, is represented by 3x3 matrix, A, and if this vector was the eigenvector of another matrix, M, with eigenvalue v, if I apply M to the matrix representation of this vector, does this holds: MA=vA? Also, if I...

hello, i have a reasearch to analyse the movement of human walking using pca. i did it like this 1. i dibide the body into some part (thigh, foot, hand, etc) 2. i film it so i can track the x position of the parts 3. i get the x to t graph for every part 4. i make a matrix which column is the...

Hi, I'm new to this forum and I couldn't find any specific sub-forum for fiber optics/waveguide theory, which my problem is regarding. Please do let me know if I should post this question some where else (and if so, where) on this forum. Anyways, here's my problem: I want to find the effective...

Hello everyone. I understand the concept of eigenvalues and eigenvectors, using usually a geometric intuition, that a eigenvectors of a matrix M are stretched by the corresponding eigenvalue, when transformed through M. My professor said that eigenvalues represent a generalization of the...

Hi. I don't understand what is meant by the eigenvalue α of a coherent state where a | α > = α | α >. The eigenket |α > is an infinite superposition of the number states , ie | α > = ∑ cn | n > and for each number state a | n > = √n | n-1 >. So for each number state the eigenvalue of the...

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?

Take the wavefunction $$e^{-i\omega t}$$ which all time dependent functions can be superposed of (right?). You can then get $$ih\frac{\partial}{\partial t}\psi=\hbar \omega \psi$$ and thus if ##\hat{E}=ih\frac{\partial}{\partial t}## then $$\hat{E}\psi=E\psi$$ What did I do wrong?

According to one of the postulates of quantum mechanics, every measured observable q is an eigenvalue of a corresponding linear Hermitian operator Q. Which means, that q must satisfy the equation Qψ = qψ. But according to Griffiths chapter 3, this equation can only be followed from σQ = 0. It...

The problem is actually of an introductory leven in Quantum Mechanics. I am doing a course on atomic and molecular physics and they wanted us to practice again some of the basics. I want to know where I went conceptually wrong because my answer doesn't give a total probability of one, which of...

Homework Statement Suppose the matrix A with real entries has the complex eigenvalue λ=α+iβ, β does not equal 0. Let Y0 be an eigenvector for λ and write Y0=Y1 +iY2 , where Y1 =(x1, y1) and Y2 =(x2, y2) have real entries. Show that Y1 and Y2 are linearly independent. [Hint: Suppose they are...

NOTE: For the answers to all these questions, I'd like an explanation (or a reference to a book or internet page) of how the answer has been derived. This question can be presumed to be for the general eigenproblem in which [ K ] & [ M ] are Hermitian matrices, with [ M ] also being positive...

I understand this question is rather marginal, but still think I might get some help here. I previously asked a question regarding the so-called computable Universe hypothesis which, roughly speaking, states that a universe, such as ours, may be (JUST IN PRINCIPLE) simulated on a large enough...

I'm going through a derivation and it shows: (dirac notation) <una|VA-AV|unb>=(anb-ana)<una|V|unb> V and A are operators that are hermition and commute with each other and ana and anb are the eigenvalues of the operator A. I imagine it is trivial and possibly doesn't even matter but why does...

There is another topic for this but I didn't quite see it and I don't know how I've gone so far through my course not asking this simple question. So what's the difference? My thought process for hydrogen. I know it can have quantised values of energy, the energy values are the Eigen values of...

Suppose ##T## is an operator in a finite dimensional complex vector space and it has a zero eigenvalue. If ##v## is the corresponding eigenvector, then $$ Tv=0v=0 $$ Does it mean then that ##\textrm{null }T## consists of all eigenvectors with the zero eigenvalue? What if ##T## does not have zero...

Homework Statement Prove that, if ##T,S\in \mathcal{L}(V)## then ##TS## and ##ST## have the same eigenvalues. Homework EquationsThe Attempt at a Solution Suppose ##T## is written in a basis in which its matrix is upper triangular, and so is ##S## (these bases may be of different list of...

Dear all, Consider the connection of two electrical circuits. Both circuits, Z1 and Z2, are stable and only one of them is non-passive. I.e., the eigenvalues are located in the LHP but Re{Z2(jw)}<0 in a frequency range. For studying the closed-loop stability, you represent the linear system by...

Question Consider the matrix $$ \left[ \matrix { 0&0&-1+i \\ 0&3&0 \\ -1-i&0&0 } \right] $$ (a) Find the eigenvalues and normalized eigenvectors of A. Denote the eigenvectors of A by |a1>, |a2>, |a3>. Any degenerate eigenvalues? (b) Show that the eigenvectors |a1>, |a2>, |a3> form an...

Suppose ##V## is a complex vector space of dimension ##n## and ##T## an operator in it. Furthermore, suppose ##v\in V##. Then I form a list of vectors in ##V##, ##(v,Tv,T^2v,\ldots,T^mv)## where ##m>n##. Due to the last inequality, the vectors in that list must be linearly dependent. This...

So, after time-independent 1D Schrodinger equation is solved, this is obtained E = n2π2ħ2/(2mL2) This means that the mass of the 'particle' is inversely related to the energy eigenvalue. Does this mean that the actual energy of the particle is inversely related to its mass? Isn't this counter...

Homework Statement there are two spheres with radius a and b that b > a.they don't have the same center and the distance between their centers is d . how can I find eigenvalue and eigenfunction of energy spacing between two spheres... I don't have any idea. please help me . Homework...

Homework Statement a particle of mass m moves in 1D potential V(x),which vanishes at infinity. Ground state eigenfunction is ψ(x) = A sech(λx), A and λ are constants. find the ground state energy eigenvalue of this system. ans: -ħ^2*λ^2/2m Homework Equations <H> =E, H = Hamiltonian. p=...

Determine the corerctions to the eigenvalue in the first approximation and the correct functions in the zeroth approximation, for a doubly degenerate level. The solution: Equation \left| V_{nn'}-E^{(1)}\delta_{nn'}\right|=0 has here the form...

So, I have the matrix: A = -1 -3 3 9 Eigenvalues i calculated to be λ = 8 and 0 Now when i calculate the Eigenvector for λ = 8, i get the answer -1 3 Then when solve for...

Hey all, I just read up on the principle of centrality, where "Think of a "network" as an NxN matrix, which has information about how N people (or N pages or N countries..) are connected to each other. Adjacency Matrix is an NxN matrix, let's say it looks something like this. People who...

How to prove that eigenvalue of lowering operator is zero?

say for example when I calculate an eigenvector for a particular eigenvalue and get something like \begin{bmatrix} 1\\ \frac{1}{3} \end{bmatrix} but the answers on the book are \begin{bmatrix} 3\\ 1 \end{bmatrix} Would my answers still be considered correct?

Suppose that we want to solve the eigenvalue equation with Dirichlet boundary conditions ## \bigg(-\frac{d^2}{dx^2}+V(x)\bigg) \phi_n = \lambda_n \phi_n,\ \ \ \ \ \ \ \ \ \ \ \ \ \phi_n(0)=0,\ \phi_n(1)=0, ## where ##0 < \lambda_1 < \lambda_2 < ...## are discrete, non-degenerate eigenvalues...