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Algebraic Multiplicity of an Eigenvalue

  1. Apr 13, 2015 #1
    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 ahead. Thank you very much for your help in this regard.

    Reference : To know how results (2) and (3) in the images come, you may please have a look at this ebook - pg 165, http://fetweb.ju.edu.jo/staff/EE/jrahhal/PDF/sc%20-%20Linear%20Algebra%20Done%20Right.pdf [Broken]

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    • 2.png
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    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. Apr 13, 2015 #2


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    In my experience, the definition you gave is of the _geometric _ multiplicity, not the algebraic one. The two multiplicities may or may not coincide.
  4. Apr 13, 2015 #3
    Thank you for the answer. The notes which I have say that the number of times λ appears on the diagonal of an upper triangular matrix is equal to dim null [T - λ I] dim V. Do you think there is error in this statement as well?"

    I have actually attempted the proof to prove this statement. Sorry about incorrectly saying it equal to algebraic multiplicity
  5. Apr 13, 2015 #4


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    Ah, sorry, I did not see the exponent dimV in your post , at first. In my experience, the geometric multiplicity of ## \lambda ## is the dimension of null## [T- \lambda I] ##. The algebraic multiplicity of ## \lambda ## is the multiplicity of the exponent of ##\lambda## in the factorization of ## A-cI ##.
    Last edited: Apr 13, 2015
  6. Apr 13, 2015 #5
    No problem :) . Do you think I made a good attempt at the proof? I just think i got stuck in the last stage of the proof
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