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. V

    Algebraic Multiplicity of an Eigenvalue

    Please have a look at the attached images.I am attempting a proof for the statement : The algebraic multiplicity of an eigen value λ is equal to dim null [T - λ I] dim V. Please advise me on how to move ahead. Apparently, I am at the final inference required for a proof but unable to move...
  2. I

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  3. R

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  4. S

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  5. H

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  6. N

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  7. J

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  8. T

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  9. G

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  10. U

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  11. N

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  12. M

    Description of eigenvector corresponding to each eigenvalue.

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  13. M

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  14. M

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  15. M

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  16. K

    Eigenvalue problem with nonlocal condition

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  17. maajdl

    Singular value decomposition and eigenvalue problem:

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  18. N

    Understanding the Eigenvalue Problem for a 4x4 Matrix with Rank 1 and Trace 10

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  19. S

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  20. C

    MHB Show V=Cx iff there are no multiple eigenvalue

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  21. D

    Eigenvalue of the system and the one of its part

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  22. L

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  23. M

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  24. H

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  25. C

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  26. T

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  27. B

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  28. K

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  31. Petrus

    MHB When there is a double root for the eigenvalue, how many eigenvectors?

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  33. V

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  34. P

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  37. A

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  38. R

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  39. J

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  40. B

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  41. M

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  43. J

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  44. R

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  45. W

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  46. W

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  47. L

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  48. T

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  49. P

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  50. J

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