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

#### lukasch

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.

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#### fresh_42

Mentor
2018 Award
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
It is a linear or quadratic equation system, depending what you mean by a decomposition.
if it is possible. So I was wonderig whether some symmetricity of the H or D can be of help.
What is $f$?
- 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.
The problem is too vaguely explained: what is known, what unknown, what is $f$, what's the goal, etc.?
(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.
Examples are a good way to start. However, you should first answer those questions above.

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