Eigenvalue Definition and 402 Threads

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. N

    I In 4x4 matrix when does row swapping not affect eigenvaluess?

    The title pretty much says it... I know that in general eigenvalues are not necessarily preserved when matrix rows or columns are swapped. But in many cases it seems they are, at least with 4x4 matrices. So is there some specific rule that says when eigenvalues are preserved if I swap two rows...
  2. T

    Eigenvalue of matrix proof by induction

    We consider base case (##n = 1##), ##B\vec x = \alpha \vec x##, this is true, so base case holds. Now consider case ##n = 2##, then ##B^2\vec x = B(B\vec x) = B(\alpha \vec x) = \alpha(B\vec x) = \alpha(\alpha \vec x) = \alpha^2 \vec x## Now consider ##n = m## case, ##B^m\vec x = B(B^{m - 1}...
  3. R

    Fourier transform of t-V model for t=0 case

    To compute the Fourier transform of the ##t-V## model for the case where ##t = 0##, we start by expressing the Hamiltonian in momentum space. Given that the hopping term ##t## vanishes, we only need to consider the potential term: $$\hat{H} = V \sum_{\langle i, j \rangle} \hat{n}_i \hat{n}_j$$...
  4. Z

    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...
  5. Z

    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...
  6. Z

    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##...
  7. M

    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!
  8. hilbert2

    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...
  9. CuriousLearner8

    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...
  10. S

    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...
  11. H

    Prove that ##\lambda## or ##-\lambda## is an eigenvalue for ##T##.

    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...
  12. J

    A Eigenvalue of the sum of two non-orthogonal (in general) ket-bras

    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...
  13. J

    I How to get the energy eigenvalue of the Hamiltonian: H0+λp/m ?

    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.
  14. M

    A First eigenvalue not matching, but all others are

    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...
  15. P

    I How to Show U|v⟩ = e^(ia)|v⟩ for Unitary Operators?

    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 =...
  16. Wannabe Physicist

    Find the eigenfunction and eigenvalues of ##\sin\frac{d}{d\phi}##

    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...
  17. M

    MHB Approximation of eigenvalue with inverse iteration method

    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...
  18. Passers_by

    Constructing an Eigenvector of S with Eigenvalue λ1

    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...
  19. B

    I It seems that the eigenvalue problem rules out the possibility of E=0?

    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...
  20. F

    Neutron quantity normalization in an eigenvalue computation

    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...
  21. Andrew1235

    Finding the directions of eigenvectors symmetric eigenvalue problem

    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...
  22. H

    Find the eigenvalues of a 3x3 matrix

    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...
  23. P

    A Eigenvalues of block matrix/Related non-linear eigenvalue problem

    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...
  24. M

    Estimating the eigenvalue with the second-order pertubation theory

    Not sure where to start!
  25. T

    I Meaning of Third Eigenvalue in a Tilted Ellipse in a 3x3 Matrix

    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...
  26. Andrei0408

    Eigenfunction proof and eigenvalue

    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?
  27. K

    Show that V is an internal direct sum of the eigenspaces

    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...
  28. K

    What can we say about the eigenvalues if ##L^2=I##?

    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...
  29. P

    I Second Matrices from Spherical Harmonics with Eigenvalue l+1

    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...
  30. P

    A Matrices from Spherical Harmonics with Eigenvalue l+1

    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\\...
  31. T

    I Simple Generalized Eigenvalue problem

    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...
  32. S

    Setting Free variables when finding eigenvectors

    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...
  33. M

    A Eigenvalue problem: locating complex eigenvalues via frequency scan

    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 +...
  34. M

    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!
  35. T

    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...
  36. K

    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...
  37. bluesky314

    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...
  38. peguerosdc

    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...
  39. M

    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...
  40. M

    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 +...
  41. V

    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...
  42. T

    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}))...
  43. S

    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...
  44. M

    Mathematica What could be causing spurious eigenvalues in my Eigensystem command?

    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...
  45. 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...
  46. O

    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...
  47. binbagsss

    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...
  48. F

    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...
  49. S

    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...
  50. renec112

    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...
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