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Often dispersion relations in plasmas are found by setting the determinant of some quantity to equal zero. My question is, how does one do this when working covariantly with tensors (e.g. Broderick, A. and Blandford, R. 2003).

Taking an example from the linked paper (Eq. 74), suppose we have some mixed rank-2 tensor $\Omega^{\mu}_{\nu}$ that we want to find the determinant of. The indices $\mu,\nu$ run $0,1,2,3$. If,

$$ \Omega^{\mu}_{\nu} = (k^{\alpha}k_{\alpha} +\omega_p^2)\delta_{\nu}^{\mu} - k^{\mu}k_{\nu}$$

then how would one find det $\Omega^{\mu}_{\nu}$?

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    $\begingroup$ see Matrix determinant lemma $\endgroup$ – AccidentalFourierTransform Feb 7 '17 at 13:22
  • $\begingroup$ Ok great - that is useful for my example given. Thanks! But what about for a more complicated expression where we have two rank 2 tensors?, e.g. $\Omega^{\mu}_{\nu} = \alpha \delta^{\mu}_{\nu} - i \gamma M^{\mu}_{\nu}$ (eq. 76) $\endgroup$ – user1887919 Feb 7 '17 at 13:29
  • $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$ – Qmechanic Feb 7 '17 at 13:38
  • $\begingroup$ Perhaps? I posted it here as it is a question I came across whilst reading a physics paper, and thought others may have similar queries from a physics perspective. $\endgroup$ – user1887919 Feb 7 '17 at 13:47

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