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. J

    Modeling the populations of foxes and rabbits given a baseline

    From solving the characteristic equations, I got that ##\lambda = 0.5 \pm 1.5i##. Since using either value yields the same answer, let ##\lambda = 0.5 - 1.5i##. Then from solving the system for the eigenvector, I get that the eigenvector is ##{i}\choose{1.5}##. Hence the complex solution is...
  2. G

    I Is there a reason eigenvalues of operators correspond to measurements?

    Given a wave function \Psi which is an eigenstate of a Hermitian operator \hat{Q}, we can measure a definite value of the observable corresponding to \hat{Q}, and the value of this observable is the eigenvalue Q of the eigenstate $$ \hat{Q}\Psi = Q\Psi $$ My question is whether it's a postulate...
  3. JD_PM

    Mathematica Is Mathematica the best option to compute the eigenvalues?

    I am wondering what's the best option to compute the eigenvalues for such a determinant$$\begin{vmatrix} \sin \Big( n \frac{\omega}{v_1} \theta \Big) & \cos \Big( n \frac{\omega}{v_1} \theta \Big) & 0 & 0 \\ 0 & 0 & \sin \Big( n \frac{\omega}{v_2} (2 \pi - \theta) \Big) & \cos \Big( n...
  4. S

    What's the formula? - eigenvectors from eigenvalues

    Anyone know what result this article is talking about? https://www.theatlantic.com/science/archive/2019/11/neutrino-oscillations-lead-striking-mathematical-discovery/602128/
  5. Greg

    MHB What is the Unexpected Discovery in Basic Math?

    A link to an interesting article I found is below: https://www.quantamagazine.org/neutrinos-lead-to-unexpected-discovery-in-basic-math-20191113/
  6. M

    A Finding eigenvalues with spectral technique: basis functions fail

    Hi PF! I'm trying to find the eigenvalues of this ODE $$y''(x) + \lambda y = 0 : u(0)=u(1)=0$$ by using the basis functions ##\phi_i = (1-x)x^i : i=1,2,3...n## and taking inner products to formulate the matrix equation $$A_{ij} = \int_0^1 \phi_i'' \phi_j \, dx\\ B_{ij} = \int_0^1...
  7. J

    Find the eigenvector with zero eigenvalues at any time t from the Hamiltonian

    I have a question relates to a 3 levels system. I have the Hamiltonian given by: H= Acos^2 bt(|1><2|+|2><1|)+Asin^2 bt(|2><3|+|3><2|) I have been asked to find that H has an eigenvector with zero eigenvalues at any time t
  8. Q

    A Eigenvalues for a non self adjoint operator

    Hi all- I am trying to obtain eigenvalues for an equation that has a very simple second order linear differential operator L acting on function y - so it looks like : L[y(n)] = Lambda (n) * y(n) Where y(n) can be written as a sum of terms in powers of x up to x^n but I find L is non self...
  9. S

    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. P

    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. QuarkDecay

    I Morse Potenials Energy eigenvalues

    I know the eigen value of energy in a Morse potential is Evib= ħωo(v+ 1/2) - ħωoxe(v+ 1/2)2 but is this the same for every Morse potential, given that the masses μ of the diatomic molecules are the same? The two potentials are these:
  12. QuasarBoy543298

    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...
  13. M

    Proving No Eigenvalues Exist for an Operator on a Continuous Function Space

    Eigenvalues ##\lambda## for some operator ##A## satisfy ##A f(x) = \lambda f(x)##. Then $$ Af(x) = \lambda f(x) \implies\\ xf(x) = \lambda f(x)\implies\\ (\lambda-x)f(x) = 0.$$ How do I then show that no eigenvalues exist? Seems obvious one doesn't exists since ##\lambda-x \neq 0## for all...
  14. R

    Find eigenvalues & eigenvectors

    Here's the problem along with the solution. The correct answer listed in the book for the eigenvectors are the expressions to the right (inside the blue box). To find the eigenvectors, I tried using a trick, which I don't remember where I saw, but said that one can quickly find eigenvectors (at...
  15. Haorong Wu

    How to diagonalize a matrix with complex eigenvalues?

    Homework Statement Diagonalize the matrix $$ \mathbf {M} = \begin{pmatrix} 1 & -\varphi /N\\ \varphi /N & 1\\ \end{pmatrix} $$ to obtain the matrix $$ \mathbf{M^{'}= SMS^{-1} }$$ Homework Equations First find the eigenvalues and eigenvectors of ##\mathbf{M}##, and then normalize the...
  16. M

    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})##...
  17. M

    MHB Interval of eigenvalues using Gershgorin circles

    Hey! :o We have the matrix $$A=\begin{pmatrix}2 & 0.4 & -0.1 & 0.3 \\ 0.3 & 3 & -0.1 & 0.2 \\ 0 & 0.7 & 3 & 1 \\ 0.2 & 0.1 & 0 & 4\end{pmatrix}$$ We get the row Gershgorin circles: $$K_1=\{z\in \mathbb{C} : |z-2|\leq 0.8 \} \\ K_2=\{z\in \mathbb{C} : |z-3|\leq 0.6 \} \\ K_3=\{z\in \mathbb{C} ...
  18. Mutatis

    Find the eigenvalues and eigenvectors

    Homework Statement Find the eigenvalues and eigenvectors of the following matrix: $$ A = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 2 \\ 0 & -1 & 0 \end{bmatrix} $$ Homework Equations Characteristic polynomial: $$ \Delta (t) = t^3 - Tr(A) t^2 + (A_{11}+A_{22} +A_{33})t - det(A) .$$ The Attempt at...
  19. Mutatis

    Find the eigenvalues and eigenvectors

    Homework Statement Find the eigenvalues and eigenvectors fro the matrix: $$ A=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} $$. Homework Equations Characteristic polynomial: ## \nabla \left( t \right) = t^2 - tr\left( A \right)t + \left| A \right|## . The Attempt at a Solution I've found...
  20. Baynie

    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...
  21. M

    A Rayleigh quotient Eigenvalues for a simple ODE

    Hi PF! Given the ODE $$f'' = -\lambda f : f(0)=f(1)=0$$ we know ##f_n = \sin (n\pi x), \lambda_n = (n\pi)^2##. Estimating eigenvalues via Rayleigh quotient implies $$\lambda_n \leq R_n \equiv -\frac{(\phi''_n,\phi_n)}{(\phi_n,\phi_n)}$$ where ##\phi_n## are the trial functions. Does the...
  22. L

    Finding <A> given the eigenvalues

    [Note from mentor: This thread was originally posted in a non-homework forum, so it lacks the homework template. Even though the solution was resolved there, the thread has been moved here for future reference.]So I'm given Φ = N(φ1+2*φ2 + 3*φ3) and the operator A with eigen values λ1 = 1, λ2...
  23. RicardoMP

    Bosonic operator eigenvalues in second quantization

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  24. CharlieCW

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  25. J

    Eigenketes and Eigenvalues of operators

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  26. Z

    Finding Eigenvalues: List of possible solutions for Lambda

    Homework Statement I got a solution for finding eigen values. It evaluates to: (Lambda)^3 -12(lambda) -16=0, Then it that the list of possible integer solutions is: +-1, +-2, +-4, _-8, +-16 (i.e. plus minus 1, plus minus 2 and so on). I can't understand, why he says list of possible solution...
  27. evinda

    MHB Eigenvalues are real numbers and satisfy inequality

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  28. Another

    How to find eigenvalues and eigenfunction

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  29. CharlieCW

    Finding Eigenvalues of an Operator with Infinite Basis

    I just began graduate school and was struggling a bit with some basic notions, so if you could give me some suggestions or point me in the right direction, I would really appreciate it. 1. Homework Statement Given an infinite base of orthonormal states in the Hilbert space...
  30. Spin One

    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...
  31. L

    A Eigenvalues and matrix entries

    If matrix has integer entries and it is hermitian, are then eigenvalues also integers? Is there some theorem for this, or some counter example?
  32. S

    Normalised eigenspinors and eigenvalues of the spin operator

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  33. karush

    MHB -Eigenvectors and Eigenvalues (311.5.5.15)

    Find a basis for eigenspace corresponding to the listed eigenvalue: just seeing if these first steps are correct \begin{align*} A_{15}&=\left[ \begin{array}{rrr} -4&1&1\\ 2&-3&2\\ 3&3&-2 \end{array} \right],\lambda=-5&(1)\\ A-(-5)i&=\left[ \begin{array}{rrr} -4&1&1\\ 2&-3&2\\ 3&3&-2 \end{array}...
  34. JTC

    Best way to learn control theory for mechanical engineers

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  35. LarryS

    I Probabilities for degenerate eigenvalues?

    In non-relativistic QM, given a wave function that has a degenerate eigenvalue for some observable, say energy. There is a whole subspace of eigenvectors associated with that single degenerate eigenvalue. How is the measurement probability for that degenerate eigenvalue computed from the...
  36. V

    Energy eigenvalues of spin Hamiltonian

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  37. LarryS

    I Eigenvectors - eigenvalues mappings in QM

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  38. P

    Eigenvectors and eigenvalues

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  39. L

    MHB Singular Values and Eigenvalues

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  40. Pushoam

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    Homework Statement Homework EquationsThe Attempt at a Solution I solved it by calculating the eigen values by ##| A- \lambda |= 0 ##. This gave me ## \lambda _1 = 6.42, \lambda _2 = 0.387, \lambda_3 = -0.806##. So, the required answer is 42.02 , option (b). Is this correct? The matrix is...
  41. astrocytosis

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    Homework Statement The Hamiltonian of a certain two-level system is: $$\hat H = \epsilon (|1 \rangle \langle 1 | - |2 \rangle \langle 2 | + |1 \rangle \langle 2 | + |2 \rangle \langle 1 |)$$ Where ##|1 \rangle, |2 \rangle## is an orthonormal basis and ##\epsilon## is a number with units of...
  42. S

    I Can a Hermitian matrix have complex eigenvalues?

    Hi, I have a matrix which gives the same determinant wether it is transposed or not, however, its eigenvalues have complex roots, and there are complex numbers in the matrix elements. Can this matrix be classified as non-Hermitian? If so, is there any other name to classify it, as it is not...
  43. Ron Burgundypants

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  44. M

    MATLAB Solving Linear System with Eigenvalues in Matlab

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  45. mr.tea

    I Eigenvalues of Circulant matrices

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  46. A

    I Eigenvalues of block matrices

    If there is matrix that is formed by blocks of 2 x 2 matrices, what will be the relation between the eigen values and vectors of that matrix and the eigen values and vectors of the sub-matrices?
  47. D

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  48. X

    I Pauli Spin Operator Eigenvalues For Two Electron System

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  49. M

    A Numerically Calculating Eigenvalues

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  50. M

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