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On the multiplicity of the eigenvalue

  1. Dec 25, 2009 #1
    On the multiplicity of the eigenvalue

    Dear friends,
    Might you tell me any hint on the multiplicity of the max eigenvalue, i.e., one, of the following matrix.

    1 0 0 0 0
    p21 0 p23 0 0
    0 p32 0 p34 0
    0 0 p43 0 p45
    0 0 0 0 0

    where, pij >0 and pi,i-1 + pi,i+1 = 1

    It is clear that the above matrix has a max eigenvalue one. Moreover, my numeric simulation shows that the multiplicity of the max eigenvalue seems to be 1. However, I failed to prove this observation. Might you help me? Thanks a lot.

    Merry Christmas
  2. jcsd
  3. Dec 28, 2009 #2
    Maxima gives me following eigenvalues for Your matrix: [itex] [[-\sqrt{p34*p43+p23*p32},\sqrt{p34*p43+p23*p32},0,1],[1,1,2,1]] [/itex] First vector are eigenvalues, second one - multiplicity.
  4. Dec 28, 2009 #3
    thank you very much for your kind help.
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