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. Come

    Eigenvalues/eignevectors of Jones matrix

    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...
  2. M

    MHB Approximation of eigenvalue with power method

    Hey! :o We have \begin{equation*}A:=\begin{pmatrix}-5.7 & -61.1 & -32.9 \\ 0.8 & 11.9 & 7.1 \\ -1.1 & -11.8 & -7.2\end{pmatrix} \ \text{ and } \ z^{(0)}:=\begin{pmatrix}1\\ 1 \\ 1\end{pmatrix}\end{equation*} I want to approximate the biggest (in absolute value) eigenvalue of $A$ with the...
  3. M

    A Eigenvalue Problem and the Calculus of Variations

    Hi PF! Given ##B u = \lambda A u## where ##A,B## are linear operators (matrices) and ##u## a function (vector) to be operated on with eigenvalue ##\lambda##, I read that the solution to this eigenvalue problem is equivalent to finding stationary values of ##(Bu,u)## subject to ##(Au,u)=1##...
  4. Drakkith

    Showing That the Eigenvalue of a Matrix is 0

    Homework Statement Show that if ##A^2## is the zero matrix, then the only eigenvalue of ##A## is 0. Homework Equations ##Ax=λx##. The Attempt at a Solution For ##A^2## to be the zero matrix it looks like: ##A^2 = AA=A[A_1, A_2, A_3, ...] = [a_{11}a_{11}+a_{12}a_{21}+a_{13}a_{31} + ... = 0...
  5. A

    MHB A question on matrix's eigenvalue problem from Eberhard Zeidler's first volume of Nonlinear Function

    The question is posted in the following post in MSE, I'll copy it here: https://math.stackexchange.com/questions/1407780/a-question-on-matrixs-eigenvalue-problem-from-eberhard-zeidlers-first-volume-o I have a question from Eberhard Zeidler's book on Non-Linear Functional Analysis, question...
  6. Drakkith

    Showing that S is an Eigenvalue of a Matrix

    Homework Statement Consider an n x n matrix A with the property that the row sums all equal the same number S. Show that S is an eigenvalue of A. [Hint: Find an eigenvector.] Homework Equations ##Ax=λx## The Attempt at a Solution S is just lambda here, so I begin solving this just like you...
  7. Drakkith

    Finding the Eigenvalue of a Matrix

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  8. A

    Eigenvalue of Exchange Operator in Hartree-Fock: 2e-

    Homework Statement Homework Equations The Attempt at a Solution [/B] ##\hat{S}_1.\hat{S}_2 = (S(S+1) - S_1(S_1 + 1) - S_2(S_2 + 1))/2## therefore singlet: ##\psi_s = \frac{\phi_a (1)\phi_b(2)(\alpha(1)\beta(2) - \alpha(2)\beta(1))}{\sqrt(2)}## So for singlet, ##\mathcal{V} = -\frac{K...
  9. N

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  10. J

    Operator in three level system -- Eigenvalues/Eigenvectors

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  11. C

    Sturm-Liouville Eigenvalue Problem (Variational Method?)

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  12. Cocoleia

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  13. Vishakha

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  14. ExplosivePete

    Power Series expansion of an eigenvalue

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  15. Mr Davis 97

    Eigenvalue Problem: Show 0 is the Only Eigenvalue of A When A^2=0

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  16. M

    Eigenvalue and Eigenvectors

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  17. M

    Finding range of bound/non bound state energies of 1D finite

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  18. A

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  19. R

    How do I find the eigenvalue given unknown rows & eigen vect

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  20. ATY

    I Complex conjugation in scalar product?

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  21. S

    I No problem, it's always good to have multiple sources!

    Hello. If I represent a vector space using matrices, for example if a 3x1 vector, V, is represented by 3x3 matrix, A, and if this vector was the eigenvector of another matrix, M, with eigenvalue v, if I apply M to the matrix representation of this vector, does this holds: MA=vA? Also, if I...
  22. martinbandung

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  23. P

    A Erroneous results when solving fiber mode eigen equation

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  24. N

    I Eigenvalue as a generalization of frequency

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  25. D

    I What is the Eigenvalue of Coherent States?

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  26. Hamza Abbasi

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  27. Isaac0427

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  28. B

    I The postulate of Quantum Mechanics and Eigenvalue equation

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  29. Smalde

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  30. Dusty912

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  31. S

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  32. ErikZorkin

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  34. S

    I Difference between expectation value and eigenvalue

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  35. M

    B Zero eigenvalue and null space

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  36. M

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  37. A

    Difference between Lyapunov and linear stability criteria

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  38. Samuel Williams

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  39. M

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  40. A

    Energy eigenvalue and mass inverse relation?

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  41. M

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  43. I

    Correction to the Eigenvalue -- doubly degenerate case

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  44. Tzabcan

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  45. N

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  46. I

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  47. C

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  48. S

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  49. S

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  50. P

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