Hi there,(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**