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.
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...
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...
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##...
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...
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...
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...
The statement " If ##T: V \to V## has the property that ##T^2## has a non-negative eigenvalue ##\lambda^2##", means that there exists an ##x## in ##V## such that ## T^2 (x) = \lambda^2 x##.
If ##T(x) = \mu x##, we've have
$$
T [T(x)]= T ( \mu x)$$
$$
T^2 (x) = \mu^2 x$$
$$
\lambda ^2 = \mu ^2...
We have a matrix ##M = \ket{\psi^{\perp}}\bra{\psi^{\perp}} + \ket{\varphi^{\perp}}\bra{\varphi^{\perp}}##
The claim is that the eigenvalues of such a matrix are ##\lambda_{\pm}= 1\pm |\bra{\psi}\ket{\varphi}|##
Can someone proof this claim? I have been told it is self-evident but I've been...
Someone says we can choose the new eigenstate: exp(-iλx/hbar)*ψ,and let the momentum operator p acts upon this new state. At the same time, so does p^2. Something miraculous will happen afterwards. My question is: how to image this point? Thank you very much.
Hi PF!
I am applying a spectral technique on a system of fluid dynamics problems. Specifically, I am looking for the characteristic frequencies, which turn out to be the eigenvalues of a matrix system ##M = \lambda K## for ##n\times n## matrices ##M,K##, which comes from a variational...
Hello,
I recently saw ##U|v\rangle= e^{ia}|v\rangle, \, a \in \mathbb{R}## and am wondering how to come up with this or how to show this.
My first thought is based on the definition of unitary operators (##UU^\dagger = I##), I would show it something like this:
##(U|v\rangle)^\dagger =...
Here is what I tried. Suppose ##f(\phi)## and ##\lambda## is the eigenfunction and eigenvalue of the given operator. That is,
$$\sin\frac{d f}{d\phi} = \lambda f$$
Differentiating once,
$$f'' \cos f' = \lambda f' = f'' \sqrt{1-\sin^2f'}$$
$$f''\sqrt{1-\lambda^2 f^2} = \lambda f'$$
I have no...
Hey! :giggle:
We have the matrix $\begin{pmatrix}2 & 1/2 & 1 \\ 1/2 & 3/2 & 1/2 \\ 1 & 1/2 & 2\end{pmatrix}$.
We take as initial approximation of $\lambda_2$ the $1.2$. We want to calculate this value approximately using the inverse iteration (2 steps) using as starting vector...
There is a eigenvector n3 of S with eigenvalue equal to λ3 and a eigenvector n1 of S with eigenvalue equal to λ1. n1 and n3 are orthogonal to each other . Construct the vector v2 so that they're orthogonal to each other(n1,v2 and n3).We can prove that v2 is an eigenvector of S . But how do we...
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...
Dear Community,
I am having a question. I have developed a simple code to perform iteration power algorithm and find the keff value of a system. However, it is not still totally clear in my mind if I have to normalize all my scores by the eigenvalue, i.e. multiply by the keff (fluxes, power...
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...
Hi,
I have a 3 mass system. ##M \neq m##
I found the forces and I get the following matrix.
I have to find ##\omega_1 , \omega_2, \omega_3## I know I have to find the values of ##\omega## where det(A) = 0, but with a 3x3 matrix it is a nightmare. I can't find the values.
I'm wondering if...
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...
While reading the Strang textbook on tilted ellipses in the form of ax^2+2bxy +cy^2=1, I got to thinking about ellipses of the form ax^2 + 2bx + 2cxy + 2dy + ey^2=1 and wondered if I could model them through 3x3 symmetric matrices. I think I figured out something that worked for xT A x, where x...
I searched through the courses but I can't find any formula to help me prove that the expression is an eigenfunction. Am I missing something? What are the formulas needed for this problem statement?
I was in an earlier problem tasked to do the same but when V = ##M_{2,2}(\mathbb R)##. Then i represented each matrix in V as a vector ##(a_{11}, a_{12}, a_{21}, a_{22})## and the operation ##L(A)## could be represented as ##L(A) = (a_{11}, a_{21}, a_{12}, a_{22})##. This method doesn't really...
This was a problem that came up in my linear algebra course so I assume the operation L is linear. Or maybe that could be derived from given information. I don't know how though. I don't quite understand how L could be represented by anything except a scalar multiplication if L...
See the first post in the previous thread ‘Matrices from Spherical Harmonics with Eigenvalue l+1’ first.
Originally when I came across the Lxyz operator and the Rlm matrices I had a different question. If this had to do with something like the quantum Hydrogen atom then why did it appear that...
I’m New to the forum. I’m Interested if a certain set of matrices have any significance. To start out the unit vectors ##\vec i , \vec j, and ~\vec k ## are replaced with two dimensional matrices.
##\sigma r = \begin{pmatrix}1&0\\0&1\\\end{pmatrix}, ~\sigma z = \begin{pmatrix}1&0\\...
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...
upon finding the eigenvalues and setting up the equations for eigenvectors, I set up the following equations.
So I took b as a free variable to solve the equation int he following way.
But I also realized that it would be possible to take a as a free variable, so I tried taking a as a free...
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 +...
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!
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...
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...
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 +...
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}))...
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 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...
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...
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...
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...
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...
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...
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...
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...