What is Eigenvalue: Definition and 400 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. M

    When can eigenvalue equations be used?

    Does anybody know what physical and mathematical conditions must be in place if we are to represent the states of physical systems using eigenvalue equations (so that we can represent the energy of a state |n> of a particle by H|n>=E|n>, etc?) Thanks!
  2. C

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    Homework Statement In quantum mechanics a physical observable is represented by an operator A. Define the terms eigenstate, eigenvalue and eigenfunction of a quantum mechanical operator. Homework Equations The Attempt at a Solution I think I know in that eq 'f' is the eigenfunction, and...
  3. M

    Know Eigenvalue and Eigenvector, How Do I Figure Out a Possible Original Matrix?

    Homework Statement This is a general question... I can easily go from a matrix A to its eigenvalues and then eigenvectors but how would I go from the eigenvalues and eigenvectors to a feasible original matrix? Any thoughts appreciated!
  4. M

    Is there more than one possibility for eigenvectors of a single eigenvalue

    I guess this is best explained with an example. The matrix (0 -1) has the eigenvalues ------------------------------------------------------------------ (1 0) i and -i. For -i we obtain ix1-x2=0 and x1+ix2=0. I got a corresponding eigen vector (1 i), but when I controlled this result with...
  5. S

    Understanding Quantum Physics: Exploring Operators and Eigenvalues

    I am new to quantum physics. Is there is any operator in quantum physics that does not show eigenvalue property? Give example please, thankx...
  6. P

    MAPLE Software Eigenvalue Problem Question

    Hi, this is probably really basic for anyone really good with MAPLE but I just solved an Eigenvalue problem in MAPLE and it displays the answer for lambda as a list since my problem contained a 6x6 matrix. My problem is that I want to be able to perform an operation of each individual output...
  7. P

    Solving Eigenvalue / Eigenvector Problem

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  8. Saladsamurai

    General Question(s) about Eigenvalue Problem

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  9. B

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  10. S

    Eigenvalue of vector space of polynomials

    Let V=C[x]10 be the fector space of polynomials over C of degree less than 10 and let D:V\rightarrowV be the linear map defined by D(f)=f' where f' denotes the derivatige. Show that D11=0 and deduce that 0 is the only eigenvalue of D. find a basis for the generalized eigenspaces V1(0), V2(0)...
  11. T

    Eigenfunction and Eigenvalue Confusion

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  12. N

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  13. M

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  14. N

    Prove or Disprove: a 3x3 matrix A can have 0 as a eigenvalue

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  15. L

    Is there a formula for calculating the derivative of the largest eigenvalue?

    Assume A=f(t) is an nxn matrix with all elements being non-negative. The eigenvalue with the largest absolute value is real and positive. We will call it r. Is there an analytical way to calculate dr/dt, or at least find the values of t for which dr/dt=0?
  16. J

    Energy Eigenvalue for a Two State System

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  17. U

    Two coupled Sturm-Liouville Eigenvalue Problems in 2-D

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  18. L

    Solve Eigenvalue Problem: q, x, A, Ak

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  19. T

    Odd result from an eigenvalue problem in the Euler equations

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  20. Z

    Product of reflection matrices without eigenvalue 1

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  21. K

    Solution to an eigenvalue problem ?

    "solution to an eigenvalue problem" ? I am trying to reproduce the results from a paper. The essence of the paper is hidden in just one equation (eq. 11) and some lines of text. For me this is going somewhat too fast. Below are the essential parts of the paper, describing the problem (and...
  22. S

    Eigenvectors with a repeated eigenvalue

    Homework Statement For the following linear system: \frac{dx}{dt} = -2x \frac{dy}{dt} = -2y Obtain the general solution. Homework Equations The Attempt at a Solution A= -2 0 0 -2 Using the determinant of A-\lambdaI I got a repeated eigenvalue of -2. I am...
  23. J

    How do I determine orthogonal eigenvectors for a repeated eigenvalue?

    Homework Statement There is an Hamiltonian operation which is given by (2 1 1) (1 2 1) = H ; 3-by-3 matrix (1 1 2) And let's have an arbtrary eigenvector (a) (b) = v ; (3x1) matrix (c) Then, from the characteristic equation, the eigenvalues are 1,4. Here eigenvalue 1 is...
  24. J

    Quantum mechanics - spin operator eigenvalue probability?

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  25. R

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  26. I

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  27. I

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  28. D

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  29. J

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  30. T

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  31. M

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  32. D

    Does anyone know any free online eigenvalue solver?

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  33. T

    Eigenvalue Factorization and Matrix Substitution

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  34. S

    Hermitian matrix with negative eigenvalue

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  35. T

    Help in Finding Eigenvectors Associated with Complet Eigenvalue

    The last matrix at the bottom of the second page is the Eigenvector found using Matlab. I'm trying to find it by hand. I found the Real Eigenvector associated with L=76.2348. But I've tried to find the Eigenvector's for the complex Eigenvalues for a while and can't get the answer given by...
  36. S

    Matrix with One Eigenvalue and Non-Eigenvector Eigenvectors

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  37. S

    Find Eigenvalue: Solutions for Beginners

    Hi all, any help greatly appreciated. Please bear in mind that i have no experience in any Physics and genuinly have no idea how to these questions. Homework Statement An eigenfunction of the operator d^2/dx^2 is ψ = e^2x. Find the corresponding eigenvalue. Homework Equations...
  38. D

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  39. Z

    Prove that a square matrix is not invertible iff 0 is an eigenvalue of A

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  40. T

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  41. S

    Does A Have a Positive Eigenvalue Using Topology?

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  42. N

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  43. K

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  44. T

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  45. J

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  46. S

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  47. W

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  48. H

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  49. N

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  50. B

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