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Properties of 4x4 symmetric matrix with eigvals E1, -E1, E2, -E2

  1. Sep 17, 2008 #1
    Hi there,

    I would appreciate if you could share your exeriences or ideas about
    properties of 4x4 symmetric/hermitean matrices H such that
    U^T H U = D = diag( E1, -E1, E2, -E2 ) or diag (E1, E2, -E1, -E2 )

    The things I would like to perform are the following
    - decompose an expression
    U tanh( D ) U^T = f(E1, -E1, E2, -E2) * H
    if it is possible. So I was wonderig whether some symmetricity of the H or D can be of help.
    - look for eigvectors - gauss elimination afer substituting known E_i is terrible,
    or I can assume some form of U, should be antisymmetric, orthogonal... but that is where I got stuck, as it depends on much parameters.
    (for 2x2 it is just U=(u v \\ -v u) with uu+vv=1, the free parameter can be
    u=(1+c)/2 v=(1-c)/2 ).

    Actually the matrix is
    H = [
    e1 d1 g 0
    d1 -e1 0 -g
    g 0 e2 d2
    0 -g d2 -e2
    But it can be rearanged in form where it is antisymetric under
    V^T H V with V = one (dir) sigma1,
    with dir I mean kronecker/direct product.

    I have an idea that similarly as for 2x2 case
    U^T sigma1 * (a b \\ b c) * sigma1 U = sigma1 diag ( E, -E ) sigma1 = - diag(E, -E)

    may it be used for 4x4 matrix as for example
    diag (E1, E2, -E1, -E2 ) = diag (E1 E2) (dir) sigma3 = ... ?
    I will think about it.

    Thanks for any note :)
  2. jcsd
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