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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.

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

Can you offer guidance or do you also need help?

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