What is Eigenvalues: Definition and 849 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. W

    Differential equations with eigenvalues.

    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...
  2. S

    Python How to Correctly Simulate an Infinite Well in Python?

    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...
  3. P

    How Can Dirac Notation Be Used to Determine Eigenvalues and Eigenfunctions?

    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)...
  4. B

    I How do i find the eigenvalues of this tough Hamiltonian?

    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...
  5. M

    B Exploring 120-Sided Dice and Eigenvalues

    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...
  6. Mr Davis 97

    Show that given n distinct eigenvalues, eigenvectors are independent

    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...
  7. Mr Davis 97

    Eigenvalues of transpose linear transformation

    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}##...
  8. J

    MHB Eigenvalues and eigenvectors

    Can anyone explain to me where I would start with this type of question please https://uploads.tapatalk-cdn.com/20170308/d5986c078504823283e8884441e39c95.jpg https://uploads.tapatalk-cdn.com/20170308/26a8c5313c1e682ae35c5b47cd2d4973.jpg
  9. H

    A Uncertainty Propagation of Complex Functions

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  10. N

    I Eigenvalues of Fermionic field operator

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  11. Sirsh

    4DOF Spur Gear System - Eigenvalues not corresponding with the Eqns?

    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...
  12. J

    I How are eigenstates and eigenvalues related in quantum mechanics?

    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...
  13. Sirsh

    I Finding eigenvalues for 4 DOF system

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  14. V

    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...
  15. TeethWhitener

    I Is a symmetric matrix with positive eigenvalues always real?

    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...
  16. J

    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...
  17. Mr Davis 97

    Show that A and its transpose have the same eigenvalues

    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| =...
  18. Mr Davis 97

    I BVP vs IVP when solving for eigenvalues

    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 =...
  19. A

    MHB Finding the Set of Eigenvalues

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  20. A

    MHB Find Eigenvalues of 4x4 Matrix with -1 & 3 Entries

    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...
  21. MrsM

    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...
  22. MrsM

    Linear Algebra: characteristic polynomials and trace

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  23. J

    A Efficiently Computing Eigenvalues of a Sparse Banded Matrix

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  24. Mr Davis 97

    T/F: Orthogonal matrix has eigenvalues +1, -1

    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...
  25. D

    I Eigenvectors for degenerate eigenvalues

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  26. H

    Eigenfunctions, eigenstates and eigenvalues

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  27. D

    I Eigenvalues, eigenvectors and the expansion theorem

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  28. L

    Oscillations: mass in the center of an octahedron -- eigenvalues?

    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...
  29. D

    How Can I Find Eigenvalues and Normalized Eigenvectors for a Matrix?

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  30. Schwarzschild90

    The eigenvalues and eigenvectors of T

    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...
  31. A

    MATLAB Calculating Total Eigenvalues in Matlab

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

    I Regarding the eigenvalues of the translation operator

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  33. D

    I Eigenvalues and Eigenvectors question

    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...
  34. R

    Understanding Eigenvalues and Eigenvectors: Exploring a Matrix Transformation

    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...
  35. I

    Eigenvalues of a spin-orbit Hamiltonian

    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...
  36. K

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    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...
  37. ShayanJ

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  38. odietrich

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

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  39. faradayscat

    Solve System with Repeated Eigenvalues

    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...
  40. faradayscat

    I Non-homogeneous systems with repeated eigenvalues

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  41. SteliosVas

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  42. P

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

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  43. Amy B

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  44. Dusty912

    Finding eigenvectors for complex eigenvalues

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  45. D

    Eigenvectors and Eigenvalues: Finding Solutions for a Matrix

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  46. E

    I Eigenspectra and Empirical Orthogonal Functions

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  47. F

    Eigenvalues of a string with fixed ends and a mass in the middle

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  48. I

    A Eigenstates of "summed" matrix

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  49. Dusty912

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  50. P

    MHB Effie's question via email about Eigenvalues, Eigenvectors and Diagonalisation

    Effie has correctly found that the eigenvalues of $\displaystyle \begin{align*} A = \left[ \begin{matrix} \phantom{-}3 & \phantom{-}2 \\ -3 & -4 \end{matrix} \right] \end{align*}$ are $\displaystyle \begin{align*} \lambda_1 = -3 \end{align*}$ and $\displaystyle \begin{align*} \lambda_2 = 2...
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