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Generalized Eigenvalue problem

  1. Mar 28, 2012 #1
    Hi all,

    I need to find the λ and the ai that solves the Generalized eigenvalue problem

    [A]{a}=-λ2 {a}

    with

    [A]=
    Code (Text):

    -1289.57,1204.12,92.5424,-7.09489,-25037.4,32022.5,-10004.3,3019.17
    1157.46,-1077.94,-0.580522,-78.9482,32022.5,-57353.5,36280.6,-10949.6
    166.577,-103.776,1494.41,-1557.21,-10004.3,36280.6,-63053.2,36776.9
    -34.4753,-22.407,-1586.37,1643.25,3019.17,-10949.6,36776.9,-28846.5
    -22840.1,29254.3,-9328.5,2914.31,-1289.57,1157.46,166.577,-34.4753
    29254.3,-51067,31724.8,-9912.15,1204.12,-1077.94,-103.776,-22.407
    -9328.5,31724.8,-54591.9,32195.6,92.5424,-0.580522,1494.41,-1586.37
    2914.31,-9912.15,32195.6,-25197.7,-7.09489,-78.9482,-1557.21,1643.25
     
    and

    =

    Code (Text):

    0,0,0,0,1875.81,0,0,0
    0,5019.07,0,0,0,22535.3,0,0
    0,0,-5019.07,0,0,0,22535.3,0
    0,0,0,0,0,0,0,937.905
    835.2,0,0,0,0,0,0,0
    0,5003.02,0,0,0,5019.07,0,0
    0,0,5003.02,0,0,0,-5019.07,0
    0,0,0,417.6,0,0,0,0
     
    Using mathematica I get for lambdas {75.1098, 35.2687, 34.3082, 15.2013, 4.3281, 1.35478,
    5.38827*10^-154, -2.06904*10^-154}

    The last two eigenvalues are zero. That made me think that two equations are redundant. I confirmed that with mathematica. Rank of A is 6.

    How can I reduce the system? I tried deleting two rows and columns, but for all the combinations the matrix B is singular.
    Notice that the first and last column/row of the matrix B have only one value different from zero. Is anything I can do to exploit that?

    Best
     
  2. jcsd
  3. Mar 28, 2012 #2

    AlephZero

    User Avatar
    Science Advisor
    Homework Helper

    Why do you think having two zero eigenvalues is an issue? If A has rank 6 and B has rank 4, that is what you would expect to get.

    For example if A was the mass matrix and B the stiffness matrix of a structure, the two zero eigenvalues just mean the system can move freely in two (generalized) directions. There is nothing "unphysical" or unreasonable about that.

    You don't say what your matrices represent, but if they come from a physics problem the zero eigenvalues probably have a physical interpretation.
     
  4. Mar 28, 2012 #3
    Thanks AlephZero. The zero eigenvalues do have a physical interpretation and it is related to what you said.

    Those special cases are treated separately and I do not need the associated eigenvectors.

    What I do up to now is solve the 8x8 system and then remove the zero eigenvalues. I want to know if there is a way reduce the system before calculating the eigenvalues so as to have the nontrivial eigenvalues.

    Best
     
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