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 :) L.