# What is Eigenvalue: Definition and 399 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. ### Direct Proof that every zero of p(T) is an eigenvalue of T

I was stuck on this problem so I looked for a solution online. I was able to reproduce the following proof after looking at the proof on the internet. By this I mean, when I wrote it below I understood every step. However, it is not a very insightful proof. At this point I did not really...
2. ### Prove 9 is eigenvalue of ##T^2\iff## 3 or -3 eigenvalue of ##T##.

Suppose ##9## is an eigenvalue of ##T^2##. Then ##T^2v=9v## for certain vectors in ##V##, namely the eigenvectors of eigenvalue ##9##. Then ##(T^2-9I)v=0## ##(T+3I)(T-3I)v=0## There seem to be different ways to go about continuing the reasoning here. My question will be about the...
3. ### Operator T, ##T^2=I##, -1 not an eigenvalue of T, prove ##T=I##.

Now, for ##v\in V##, ##(T+I)v=0\implies Tv=-v##. That is, the null space of ##T+I## is formed by eigenvectors of ##T## of eigenvalue ##-1##. By assumption, there are no such eigenvectors (since ##-1## is not an eigenvalue of ##T##). Hence, if ##(T-I)v \neq 0## then ##(T+I)(T-I)v\neq 0##...
4. ### Understanding Eigenvalues of a Matrix

For this, I am confused by the second line. Does someone please know how it can it be true since the matrix dose not have an inverse. Many thanks!
5. ### A The eigenvalue power method for quantum problems

The classical "power method" for solving one special eigenvalue of an operator works, in a finite-dimensional vector space, as follows: suppose an operator ##\hat{A}## can be written as an ##n\times n## matrix, and its unknown eigenvectors are (in Dirac bra-ket notation) ##\left|\psi_1...
6. ### A Eigenvalue Problem of Quantum Mechanics

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...
7. ### Determine eigenvalue-problem for steel pole

If we assume that ##\psi## has a Fourier transform ##\hat{\psi}##, so that ##\psi(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat{\psi}(x,\omega)e^{i\omega t}\mathrm{d}\omega##, then the wave equation reduces to ##-\rho\omega^2\hat{\psi}(x,\omega)=E\frac{\partial^2 \hat{\psi}(x,\omega)}{\partial...

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### I Know a simple, linear, complex, eigenvalue BVP?

Hi PF! I'm trying to find a 1D, linear, complex, 2nd order, eigenvalue BVP: know any that admit analytic solutions? Can't think of any off the top of my head. Thanks!
32. ### I Eigenvalue Problem -- Justification

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...
33. ### I Find Practical Resonance Frequencies in Linear Differential Equations

Hi all, I would like to know what is the equation upon which I can use to determine the practical resonance frequencies in a system of second order, linear differential equations. First some definitions: What I mean by practical resonance frequencies, is the frequencies that a second order...
34. ### I Question about an eigenvalue problem: range space

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...
35. ### I Confusion with Dirac notation in the eigenvalue problem

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...
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### Quadratic eigenvalue problem and solution (solved in Mathematica)

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...
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### MATLAB Solving Polynomial Eigenvalue Problem

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 +...
38. ### How to find the diagonal matrix and it's dominant eigenvalue

Homework Statement Consider the following vectors, which you can copy and paste directly into Matlab. x = [2 2 4 6 1 5 5 2 6 2 2]; y = [3 3 3 6 3 6 3 2 3 2]; Use the vectors x and y to create the following matrix. 2 3 0 0 0 0 0 0 0 0 0 3 2 3 0 0 0 0 0 0 0 0 0 3 4 3 0 0 0 0 0 0 0 0 0 3 6 6 0...
39. ### I Calculating the eigenvalue of orbital angular momentum

Hello, I'm trying to calculate the measurement of the orbital angular momentum of the state l=1 and m = -1. The operator for the angular momentum squared is ## L^2 = -\hbar (\frac{1}{sin\theta}(\frac{\partial}{\partial \theta}(sin\theta\frac{\partial}{\partial \theta}))...
40. ### Eigenvalue problem -- Elastic deformation of a membrane

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...
41. M

### Mathematica Solving eigenvalue problems

Hi PF! I have an eigenvalue problem ##K = \lambda M##. Matrices ##M,K## are constructed via integrating combinations of basis functions (similar to a finite element method). The system is the size of the number of basis functions included: if we choose ##3##, the first three basis functions...
42. M

### Mathematica Eigenvalue problem and badly conditioned matrices

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...
43. ### Show that eigenvalue of A + eigvalueof B ≠ eigvalue of A+B?

Homework Statement Let A and B be nxn matrices with Eigen values λ and μ, respectively. a) Give an example to show that λ+μ doesn't have to be an Eigen value of A+B b) Give an example to show that λμ doesn't have to be an Eigen value of AB Homework Equations det(λI - A)=0 The Attempt at a...
44. ### Linear Algebra: 2 eigenfunctions, one with eigenvalue zero

Homework Statement If I have two eigenfunctions of some operator, that are linearly indepdendent e.g ##F(x) , G(x)+16F(x) ## and ##F(x)## has eigenvalue ##0##, does this mean that ## G(x) ## must itself be an eigenfunction? I thought for sure yes, but the way I particular question I just...
45. ### Calculating eigenvectors/values from Hamiltonian

Homework Statement I've constructed a 3D grid of n points in each direction (x, y, z; cube) and calculated the potential at each point. For reference, the potential roughly looks like the harmonic oscillator: V≈r2+V0, referenced from the center of the cube. I'm then constructing the Hamiltonian...
46. ### Normalised eigenspinors and eigenvalues of the spin operator

Homework Statement Find the normalised eigenspinors and eigenvalues of the spin operator Sy for a spin 1⁄2 particle If X+ and X- represent the normalised eigenspinors of the operator Sy, show that X+ and X- are orthogonal. Homework Equations det | Sy - λI | = 0 Sy = ## ħ/2 \begin{bmatrix} 0...
47. ### First order pertubation of L_y operator

Hi, I am trying to solve an exam question i failed. It's abput pertubation of hydrogen. I am given the following information: The matrix representation of L_y is given by: L_y = \frac{i \hbar}{\sqrt{2}} \left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & -1...
48. ### MHB Convergence of iteration method - Relation between norm and eigenvalue

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...
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### A Solving an ODE Eigenvalue Problem via the Ritz method

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) =...
50. ### Eigenvalues/eignevectors of Jones matrix

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...