Calculate Eigenvalues of Electromagnetic & Stress-Energy Tensors

[ergospherical]

How can we (as nicely as possible... i.e. not via characteristic polynomial) calculate the eigenvalues of $F_{ab} = \partial_a A_b -\partial_b A_a$ and $T_{ab} = F_{ac} {F_b}^c- (1/4) \eta_{ab} F^2$ and what is their physical meaning?

 

[robphy]

Useful fact:
The eigenvalues of a real antisymmetric matrix are imaginary. (So, maybe compute the eigenvalues of its square.)
The eigenvalues of a real symmetric matrix are real.

Related: https://www.physicsforums.com/threads/em-tensor-invariants.676181/post-4295385
Possibly useful: https://en.wikipedia.org/wiki/Classification_of_electromagnetic_fields

 

[Orodruin]

I mean, first of all you would have to define what you mean by "eigenvalues". The entire concept of an eigenvalue is that you have an operator from a vector space to itself. As such, neither $T_{\mu\nu}$ or $F_{\mu\nu}$ have eigenvalues because they are (0,2) tensors. You can, of course, raise and lower an index using the metric if you have one, but then you no longer have a symmetric or anti-symmetric matrix.

 

[ergospherical]

Possibly useful: https://en.wikipedia.org/wiki/Classification_of_electromagnetic_fields

Useful, thanks.