Eigenvalues Definition and 820 Threads

If there is matrix that is formed by blocks of 2 x 2 matrices, what will be the relation between the eigen values and vectors of that matrix and the eigen values and vectors of the sub-matrices?

A fictitious system having three degenerate angular momentum states with ##\ell=1## is described by the Hamiltonian \hat H=\alpha (\hat L^2_++\hat L^2_-) where ##\alpha## is some positive constant. How to find the energy eigenvalues of ##\hat H##?

I'm studying for a qualifying exam and I see something very strange in the answer key to one of the problems from a past qualifying exam. It appears the sigma^2 for a two electron system has eigenvalues according to the picture below of 4s(s+1) while from my understand of Sakurai it should have...

Hi PF! I am trying to solve the eigenvalue problem ##Ax = \lambda Bx## where I have numerical entries for the square matrices ##A## and ##B##. I solve this by taking $$Ax = \lambda Bx\implies\\ B^{-1}Ax- \lambda Ix=0\implies\\ (B^{-1}A-\lambda I)x=0$$ where I then use a built in function in...

Homework Statement Given $$u''(x)+\lambda u = 0\\ u(-1)=u(1)=0.$$ If ##\lambda_0## is the lowest eigenvalue, show that ##4 \lambda_0 = \pi^2##. Homework Equations $$\lambda_0 = glb\frac{(L(u),u)}{(u,u)}$$ where ##glb## denotes greatest lower bound and ##L## is the Sturm-Louiville operator. I...

Homework Statement Find all solutions of the given differential equations: ## \frac{dx}{dt} = \begin{bmatrix} 6 & -3 \\ 2 & 1 \end{bmatrix} x ## Homework EquationsThe Attempt at a Solution So, we just take the determinate of A-I##\lambda## and set it equal to 0 to get the eigenvalues of 3...

I'm trying to get the eigenfunctions and eigenvalues (energies) of an infinite well in Python, but I have a few things I can't seem to fix or don't understand... Here's the code I have: from numpy import * from numpy.linalg import eigh import matplotlib.pyplot as plt from __future__ import...

Homework Statement I have the following question (see below) Homework Equations The eigenvalue equation is Au = pu where u denotes the eigenstate and p denotes the eigenvalue The Attempt at a Solution I think that the eigenvalues are +1 and - 1, and the states are (phi + Bphi) and (phi-Bphi)...

I have this Hamiltonian --> (http://imgur.com/a/lpxCz) Where each G is a matrix. I want to find the eigenvalues but I'm getting hung up on the fact that there are 6 indices. Each G matrix lives in a different space so I can't just multiply the G matrices together. If I built this Hamiltonain...

Here is a picture of a set of 120 sided dice. Each die has 120 eigenvalues. It is easy to see that as the number of eigenvalues increases, the probability of any eigenvalue gets smaller. In the limit where the number of eigenvalues is ##\infty## the probability of anyone eigenvalue approaches...

Homework Statement Let ##T## be a linear operator on a vector space ##V##, and let ##\lambda_1,\lambda_2, \dots, \lambda_n## be distinct eigenvalues of ##T##. If ##v_1, v_2, \dots , v_n## are eigenvectors of ##T## such that ##\lambda_i## corresponds to ##v_i \ (1 \le i \le k)##, then ##\{ v_1...

Homework Statement If ##A## is an ##n \times n## matrix, show that the eigenvalues of ##T(A) = A^{t}## are ##\lambda = \pm 1## Homework EquationsThe Attempt at a Solution First I assume that a matrix ##M## is an eigenvector of ##T##. So ##T(M) = \lambda M## for some ##\lambda \in \mathbb{R}##...

Suppose I have some observables \alpha, \beta, \gamma whose central values and uncertainties \sigma_{\alpha}, \sigma_{\beta}, \sigma_{\gamma} are known. Define a function f(\alpha, \beta, \gamma) which has both real and complex parts. How do I do standard error propagation when imaginary...

Hello, I'm a bit confused about the eigenvalues of the second quantized fermionic field operators \psi(x)_a. Since these operators satisfy the condition \{\psi(x)_a, \psi(y)_b\} = 0 the eigenvalues should also anti-commute? Does this mean that the eigenvalues of \psi(x)_a are...

Hi there, I am modelling a four degree of freedom system which is the dynamics of two spur gears in mesh, having two rotational and two translation degrees of freedom, respectively, a diagram exhibiting the system can be seen below. I have derived the equations of motion (EOM) and...

Hi, I have come across two definitions of eigenstates (and eigenvalues), both of which I understand but I don't understand how the two are related: 1) An eigenstate is one where you get the original function back, usually with some multiple, which is called the eigenvalue. 2) An eigenstate...

Hey all, I've derived a fourth order dynamic system as represented by the following: I need to determine the eigenvalues for this system to check whether they're purely real with no imaginary components. How should I go about doing this? I have done eigenvalue problems in the past, but not to...

Homework Statement Consider a particle with angular momentum l=1. Write down the matrix representation for the operators L_x,\,L_y,\,L_z,for this particle. Let the Hamiltonian of this particle be H = aL\cdot L-gL_z,\,g>0.Find its energy values and eigenstates. At time t=0,we make a measurement...

I split off this question from the thread here: https://www.physicsforums.com/threads/error-in-landau-lifshitz-mechanics.901356/ In that thread, I was told that a symmetric matrix ##\mathbf{A}## with real positive definite eigenvalues ##\{\lambda_i\} \in \mathbb{R}^+## is always real. I feel...

I am using arpack (the dsdrv1 driver) to iteratively solve the eigenvalue problem Ax = λx I am interested in the first m eigenvectors, and I have very good initial approximations for these vectors, so I would like to use my m starting vectors as an initial guess. However...

Homework Statement Show that ##A## and ##A^T## have the same eigenvalues. Homework EquationsThe Attempt at a Solution If they have the same eigenvalues, then ##Ax = \lambda x## iff ##A^T x = \lambda x##. In other words, we have to show that ##|A - \lambda I| = 0## iff ##|A^T - \lambda I| =...

I am being asked to find ##\lambda## such that ##y'' + \lambda y = 0; ~y(0) =0; ~y'( \pi ) = 0##. This is an eigenvalue problem where we are given boundary conditions. ##\lambda## can be found such that we don't have a trivial solution if we test the different cases when ##\lambda < 0, \lambda =...

How do I find the set of all eigenvalues of $F^2$, if the problem gives me that $F: X \mapsto X$ is the derivative mapping defined in the space $X$ of all infinitely differentiable real functions? I found this solution from an online source which is not known for being reliable. Therefore I am...

Could anyone help me figure out the eigenvalues of a $4 \times 4$ matrix, with all entries being $-1$ except the entries of non-main diagonal being $3$? $$ A = \begin{pmatrix} 3 &-1 &-1 &-1\\ -1 &3 &-1 &-1\\ -1 &-1 &3 &-1\\ -1 &-1 &-1 &3\\ \end{pmatrix}$$ For small matrices, I would use...

Homework Statement Homework Equations determinant is the product of the eigenvalues... so -1.1*2.3 = -2.53 det(a−1) = 1 / det(A), = (1/-2.53) =-.3952 The Attempt at a Solution If it's asking for a quality of its inverse, it must be invertible. I did what I showed above, but my answer was...

The question is : Is it true that two matrices with the same characteristic polynomials have the same trace? I know that similar matrices have the same trace because they share the same eigenvalues, and I know that if two matrices have the same eigenvalues, they have the same trace. But I am...

I have a Hamiltonian represented by a penta-diagonal matrix The first bands are directly adjascent to the diagonals. The other two bands are N places above and below the diagonal. Can anyone recommend an efficient algorithm or routine for finding the eigenvalues and eigenvectors?

Homework Statement If a 3 x 3 matrix A is diagonalizable with eigenvalues -1, and +1, then it is an orthogonal matrix. Homework EquationsThe Attempt at a Solution I feel like this question is false, since the true statement is that if a matrix A is orthogonal, then it has a determinant of +1...

I am looking at some notes on Linear algebra written for maths students mainly to improve my Quantum Mechanics. I came across the following example - $$ \begin{pmatrix} 2 & -3 & 1 \\ 1 & -2 & 1 \\ 1 & -3 & 2 \end{pmatrix} $$ The example then gives the eigenvalues as 0 and 1(doubly degenerate)...

Homework Statement The problem states consider A_hat=exp(b*(d/dx)). Then says ψ(x) is an eigenstate of A_hat with eigenvalue λ, then what kind of x dependence does the function ψ(x) have as x increases by b,2b,...? Homework EquationsThe Attempt at a Solution Started out by doing...

If i have an arbitrary ket then i know it can always be expressed as a linear combination of the basis kets.I now have an operator A which has 2 eigenvalues +1 and -1. The corresponding eigenvectors are | v >+ = k | b > + m | a > and | v >- = n | c > where | a > , | b > and | c > are...

You have an infinitesimally small mass in the center of octahedron. Mass is connected with 6 different springs (k_1, k_2, ... k_6) to corners of octahedron. Equilibrium position is in the center, you don't take into account gravity, only springs. Find normal modes and frequencies. Relevant...

Homework Statement Find the eigen values and normalized eigen vectors for the matrix cosθ sinθ -sinθ cosθ 2. The attempt at a solution Well I did the eigen values hope they are correct but can't solve for eigen vectors Eigen values are λ = cosθ ± isinθ on solving for eigen vector for...

Homework Statement Homework Equations The lattice laplacian is defined as \Delta^2 = \frac{T}{\tau} , where T is the transition matrix \left[ \begin{array}{cccc} -2 & 1 & 0 & 0 \\ 1 & -2 & 1 & 0 \\ 0 & 1 & -2 & 1 \\ 0 & 0 & 1 & -2 \end{array} \right] and \tau is a time constant, which is...

Hi. How do you calculate Total eigenvalues smaller than a certain value in Matlab?

I don't understand what the eigenvalue of a translation operator means physically. The eigenvalues of other operators like momentum and hamiltonian give us the physically measurable values I suppose. Then what exactly do we obtain by the translation eigenvalues? I am new to the field of quantum...

Let's say i have the 3x3 matrix a 0 0 b 0 0 1 2 1 it's eigenvalues are e1 =a, e2 = 0, e3 = 1. now if a = / = 0, 1 i have 3 distinct eigenvalues and the matrix is surely can be Diagonalizable. so if i try to solve for the eigenvector for the eigenvalue e1 =a: 0 0 0 b -a 0 1 2 1-a...

Homework Statement \frac{d\vec{Y}}{dt} = \begin{bmatrix} 2 & 4 \\ 3 & 6 \end{bmatrix} \vec{Y} Find the eigenvalues and eigenvectors Homework EquationsThe Attempt at a Solution I found the eigenvectors to be \vec{v_1} = \begin{bmatrix} 2 \\ 1 \end{bmatrix} , \vec{v_2} = \begin{bmatrix} 2...

Good day everyone, The question is as following: Consider an electron gas with Hamiltonian: \mathcal{H} = -\frac{\hbar^2 \nabla^2}{2m} + \alpha (\boldsymbol{\sigma} \cdot \nabla) where α parameterizes a model spin-orbit interaction. Compute the eigenvalues and eigenvectors of wave vector k...

Homework Statement we have this matrix 6 - 1 0 -1 -1 -1 0 -1 1 We need to find it's eigenvalues and eigenvectors Homework Equations The Attempt at a Solution[/B] I wrote the characteristic equation - det(A- λxunit matrix) to find the roots and got (-λ^3)+8(λ^2)+λ-6 instead of...

How can I prove that the eigenvalues of the operator ## i\gamma^\mu \partial_\mu ## are non-negative? I've tried using the ansatz ## \psi=u(p) e^{ip_\nu x^\nu} ## but it didn't help. I've also tried playing with the equation using the properties of gamma matrices but that doesn't seem to lead...

I'm looking for the general form of a symmetric 3×3 matrix (or tensor) ##\textbf{A}## with only two different eigenvalues, i.e. of a matrix with the diagonalized form ##\textbf{D}=\begin{pmatrix}a& 0 & 0\\0 & b & 0\\0 & 0 & b\end{pmatrix} = \text{diag}(a,b,b)##. In general, such a matrix can be...

Homework Statement I want to solve this systemx' = \left( \begin{array}\\ 7 & 1 \\ -4 & 3 \end{array} \right)x + \left( \begin{array}\\ t \\ 2t \end{array} \right) Homework EquationsThe Attempt at a Solution i found the eigenvalues to both be 5. The eigenvector is (1,-2) and the generalized...

Quick question, can you solve non-homogeneous systems with repeated eigenvalues the same ways? i.e. variation of parameters, undetermined coefficients, etc... would the fundamental matrix contain the solution with the generalized eigenvalue? Thanks!

Homework Statement Okay this is the problem it seems so easy but i just cannot for the life of me get it to click into my mind, I have 4 unknowns and 5 equations and i have to put it into a matrix and try solve it matricies or eigenvalues/eigenvectors. The 5 equations are: a= b/2 b=a/3 + d...

This question was inspired by 3c) on https://people.phys.ethz.ch/~muellrom/qm1_2012/Solutions4.pdf Given the operator \hat{B} = \left(\matrix{b&0&0\\0&0&-ib\\0&ib&0}\right) I find correctly that the eigenvalues are \lambda = b, \pm b. To find the eigenvectors for b, I do the following...

is exp (-kx) an eigenfunction?

Homework Statement So I have been having trouble with finding the proper eigen vector for a complex eigen value for the matrix A=(-3 -5) . .....(3 1) had a little trouble with formating this matrix sorry The eigen values are -1+i√11 and -1-i√11 The Attempt at a Solution using AY-λY=0...

Homework Statement Find the eigenvalues and associated eigenvector of the following matrix: Homework EquationsThe Attempt at a Solution We have a theorem in our lectures notes that states that if a matrix is invertible the only eigenvector in its kernel will be the zero vector. In order...

Are the Eigenspectra (a spectrum of eigenvalues) and the Empirical Orthogonal Functions (EOFs) the same? I have known that both can be calculated through the Singular Value Decomposition (SVD) method. Thank you in advance.