I Calculate Eigenvalues of Electromagnetic & Stress-Energy Tensors

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Calculating the eigenvalues of the electromagnetic tensor ##F_{ab}## and the stress-energy tensor ##T_{ab}## can be approached without using the characteristic polynomial. The eigenvalues of the antisymmetric matrix ##F_{ab}## are imaginary, suggesting that computing the eigenvalues of its square may be a viable method. In contrast, the eigenvalues of the symmetric matrix ##T_{ab}## are real. However, the discussion highlights that the tensors themselves do not have eigenvalues in the traditional sense, as they are (0,2) tensors rather than operators on a vector space. Understanding the physical meaning of these eigenvalues requires further clarification of the definitions and context of the tensors involved.
ergospherical
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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?
 
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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.
 
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Moderator's note: Spin-off from another thread due to topic change. In the second link referenced, there is a claim about a physical interpretation of frame field. Consider a family of observers whose worldlines fill a region of spacetime. Each of them carries a clock and a set of mutually orthogonal rulers. Each observer points in the (timelike) direction defined by its worldline's tangent at any given event along it. What about the rulers each of them carries ? My interpretation: each...

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