# Eigenvalues Definition and 52 Discussions

In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by

λ

{\displaystyle \lambda }
, is the factor by which the eigenvector is scaled.
Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated.

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1. ### If |a> is an eigenvector of A, is f(B)|a> an eigenvector of A?

Hi, If ##|a\rangle## is an eigenvector of the operator ##A##, I know that for any scalar ##c \neq 0## , ##c|a\rangle## is also an eigenvector of ##A## Now, is the ket ##F(B)|a\rangle## an eigenvector of ##A##? Where ##B## is an operator and ##F(B)## a function of ##B##. Is there way to show...
2. ### FEM, Matlab and the modes of an element

Hello I wrote a Matlab code to form the 8 by 8 stiffness matrix of a single, 4-noded element, for a plane strain problem for an isotropic element. I conduct an eigenvalue analysis on this matrix Matlabe reports 5 non-zero eigenvalue modes, and 3 zero-eigenvalue modes (as expected) Of the 3...

9. ### I Find matrix of linear transformation and show it's diagonalizable

The strategy here would probably be to find the matrix of ##F##. How would one go about doing that? Since ##V## is finite dimensional, it must have a basis...
10. ### I Measurement with respect to the observable Y

Hello, I would like to start with an assumption. Suppose a system is in the state: $$|\psi\rangle=\frac{1}{\sqrt{6}}|0\rangle+\sqrt{\frac{5}{6}}|1\rangle$$ The question is now: A measurement is made with respect to the observable Y. The expectation or average value is to calculate. My first...
11. ### I Applying an observable operator on the current state

hey :) assume I have an operator A with |ai> eigenstates and matching ai eigenvalues, and assume my system is in state |Ψ> = Σci|ai> I know that applying the measurement that corresponds to A will collapse the system into one of the |ai>'s with probability |<Ψ|ai>|2. with that being...
12. ### Poincare algebra and its eigenvalues for spinors

Homework Statement Show that for $$W^\mu = -\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}M^{\nu\rho}P^{\sigma},$$ where ##M^{\mu\nu}## satisfies the commutation relations of the Lorentz group and ##\Psi## is a bispinor that transforms according to the ##(\frac{1}{2},0)\oplus(0,\frac{1}{2})##...
13. ### MATLAB Code: Stationary Schrodinger EQ, E Spec, Eigenvalues

Hello everyone, For weeks I have been struggling with this quantum mechanics homework involving writing a code to determine the energy spectrum and eigenvalues for the stationary Schrodinger equation for the harmonic oscillator. I can't find any resources anywhere. If anyone could help me get...
14. ### I Eigenkets belonging to a range of eigenvalues

When one wants to represent a general ket in a basis consisting of eigenkets each attributed to an eigenvalue in a range, say from a to b, why does one take the integral of said kets from a to b w.r.t. the eigenvalues? I understand that the integral here plays a role analogous to a sum in the...
15. ### Best way to learn control theory for mechanical engineers

Hello, I have begun to teach myself Control Theory. I am looking for a book that is focused for mechanical engineers. I do not mind examples in electrical engineering, but they bore me (no offense). Also, I find some books begin with Laplace Transforms. Yet I found this online lecture...
16. ### Eigenvalues and vectors of a 4 by 4 matrix

Homework Statement Coupled Harmonic Oscillators. In this series of exercises you are asked to generalize the material on harmonic oscillators in Section 6.2 to the case where the oscillators are coupled. Suppose there are two masses m1 and m2 attached to springs and walls as shown in Figure...
17. ### I Eigenvalues of Circulant matrices

Hi, I am studying about circulant matrices, and I have seen that one of the properties of such matrices is the eigenvalues which some combinations of roots of unity. I am trying to understand why it is like that. In all the places I have searched they just show that it is true, but I would like...
18. D

### To find the energy eigenvalues in the 3D Hilbert space

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##?
19. ### Python Simulating an infinite well

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...
20. ### A Uncertainty Propagation of Complex Functions

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...
21. ### Eigenvectors of Ly and associated energies

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...
22. ### A How do I supply arpack drivers with all starting vectors?

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...
23. ### Using eigenvalues to get determinant of an inverse matrix

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...
24. ### Linear Algebra: characteristic polynomials and trace

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...
25. ### I General form of symmetric 3x3 matrix with only 2 eigenvalues

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...
26. ### Repeated eigenvalues

Homework Statement I want to solve this system x' = \left( \begin{array}\\ 7 & 1 \\ -4 & 3 \end{array} \right)x + \left( \begin{array}\\ t \\ 2t \end{array} \right) Homework Equations The Attempt at a Solution i found the eigenvalues to both be 5. The eigenvector is (1,-2) and the...
27. ### I Non-homogeneous systems with repeated eigenvalues

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!
28. ### System of equations

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...
29. ### I When should one eigenvector be split into two (same span)?

This question was inspired by 3c) on https://people.phys.ethz.ch/~muellrom/qm1_2012/Solutions4.pdf [Broken] 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...
30. ### I Eigenspectra and Empirical Orthogonal Functions

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.
31. ### A Eigenstates of "summed" matrix

Hi to all. Say that you have an eigenvalue problem of a Hermitian matrix ##A## and want (for many reasons) to calculate the eigenvalues and eigenstates for many cases where only the diagonal elements are changed in each case. Say the common eigenvalue problem is ##Ax=λx##. The ##A## matrix is...
32. ### Energy eigenvalue and mass inverse relation?

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...
33. ### Why are the eigenvectors the axes of an ellipse?

I'm almost there in terms of understanding it, but I need to go beyond the text. Here is the example problem: imgur link: http://i.imgur.com/UMj55tF.jpg I can see that where we have 1 = \vec{x}^T A \vec{x} = \lambda \vec{x}^T \vec{x} that 1=\lambda \vec{x}^T \vec{x} = \lambda ||\vec{x}||^2...
34. ### How can e^{Diag Matrix} not be an infinite series?

So, in a section on applying Eigenvectors to Differential Equations (what a jump in the learning curve), I've encountered e^{At} \vec{u}(0) = \vec{u}(t) as a solution to certain differential equations, if we are considering the trial substitution y = e^{\lambda t} and solving for constant...
35. ### Matrix with repeated eigenvalues is diagonalizable....?

MIT OCW 18.06 Intro to Linear Algebra 4th edt Gilbert Strang Ch6.2 - the textbook emphasized that "matrices that have repeated eigenvalues are not diagonalizable". imgur: http://i.imgur.com/Q4pbi33.jpg and imgur: http://i.imgur.com/RSOmS2o.jpg Upon rereading...I do see the possibility...
36. ### Find eigenvalues and eigenvectors of weird matrix

Homework Statement find eigenvalues and eigenvectors for the following matrix |a 1 0| |1 a 1| |0 1 a| Homework Equations The Attempt at a Solution I'm trying to find eigenvalues, in doing so I've come to a dead end at 1 + (a^3 - lambda a^2 -2a^2 lambda + 2a lambda^2 + lambda^2 a - lambda^3...
37. ### Eigenvalues of disturbed Hamiltonian

Hello everyone! I'm trying to follow a solution to a problem from the book "Problems and Solutions on Quantum Mechanics", it's problem 1017. There's a step where they go on too fast, and I can't follow. I've posted the solution and where my problem is down below. Homework Statement The dynamics...
38. ### Eigenvalues are invariant but eigenvectors are not

Hi there. How would I show that the eigenvalues of a matrix are an invariant, that is, that they depend only on the linear function the matrix represents and not on the choice of basis vectors. Show also that the eigenvectors of a matrix are not an invariant. Explain why the dependence of the...
39. ### Eigenvalue and vector quick question

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...
40. ### LAPACK dgeev: parameter had illegal value

Mod note: I revised the code below slightly, changing the loop control variable i to either j or k. The reason for this is that the browser mistakes the letter i in brackets for the BBCode italics tag, which causes some array expressions to partially disappear. Hello, I am trying for the first...
41. ### Why does this shortcut for eigenvectors of 2x2 symmetric work?

Hi, I'k looking at some MATLAB code specifically eig2image.m at: http://www.mathworks.com/matlabcentral/fileexchange/24409-hessian-based-frangi-vesselness-filter/content/FrangiFilter2D So, I understand how the computations are done with respect to the eigenvector / eigenvalues and using...
42. ### Elementary quantum spin in Sakurai

Homework Statement I am currently working on a seemingly straightforward eigenvalue problem appearing as problem 1.8 in Sakurai's Modern QM. He asks us to find an eigenket \vert\vec S\cdot\hat n;+\rangle with \vec S\cdot\hat n\vert\vec S\cdot\hat n;+\rangle = \frac\hbar 2\vert\vec S\cdot\hat...
43. ### Numerical solution to Schrödinger equation - eigenvalues

Not sure whether to post this here or in QM: I trying to numerically solve the Schrödinger equation for the Woods-Saxon Potential and find the energy eigenvalues and eigenfucnctions but I am confused about how exactly the eigenvalues come about. I've solved some differential equations in the...
44. ### The Cooper pair box Hamiltonian in the matrix form

Hello, In my problem I need to We are advised to create the Cooper pair box Hamiltonian in a matrix form in the charge basis for charge states from 0 to 5. Here is the Hamiltonian we are given H=E_C(n-n_g)^2 \left|n\right\rangle\left\langle...
45. ### Energy Eigenvalues for Ion with Spin

Homework Statement An ion has effective spin ħ. The spin interacts with a surrounding lattice so that: Hspin = A S2 z. I first had to write H as a matrix. Then i had to find the energy eigenvalues. Homework Equations The Attempt at a Solution I figured j=1 and mj = 1,0,-1 S2 z = ħ2(1 0...
46. ### Little question about eigenvalues

Hey there, I'm thinking about if one of the eigenvalues is zero (means determinant is 0. right?) So, is there any possibility to non-zero eigenvalue also exists?
47. ### Finding the unit vector for an ellipse

Homework Statement Given the ellipse ##0.084x^2 − 0.079xy + 0.107y^2 = 1 ## Find the semi-major and semi-minor axes of this ellipse, and a unit vector in the direction of each axis. I have calculated the semi-major and minor axes, I am just stuck on the final part. Homework Equations this...
48. ### Abstract Linear Algebra: Eigenvalues & Eigenvectors

Homework Statement Let V be a finite dimensional vector space over ℂ . Show that any linear transformation T:V→V has at least one eigenvalue λ and an associated eigenvector v. Homework Equations The Attempt at a Solution Hey everyone I've been doing sample questions in the build up to an...
49. ### Differential Equations System Solutions

Homework Statement Consider the initial value problem for the system of first-order differential equations y_1' = -2y_2+1, y_1(0)=2 y_2' = -8y_1+2, y_2(0)=-1 If the matrix [ 0 -2 -8 0 ] has eigenvalues and eigenvectors L_1= -4 V_1= [ 1...
50. ### Expecting the possible event of zero probability

Consider a potential well in 1 dimension defined by $$V(x)= \begin{cases} +\infty &\text{if}& x<0 \text{ and } x>L\\ 0 &\text{if} &0\leq x\leq L \end{cases}$$ The probability to find the particle at any particular point x is zero. P(\{x\}) = \int_S \rho(x)\mathrm{d}x=0 ;\forall\; x \in...