I encounter some problems on the properties of eigenvalues of anti-symmetric matrices $F_{\mu \nu}$. If we are working in Lorentz signature, the metric as $\eta_{\mu\nu} = \{-1,1,1,... \}$ and the inner product of two vector $u^{\mu}$ and $v^{\mu}$ is \begin{equation} \langle u,v \rangle=u^{\mu}v^{\nu}\eta_{\mu \nu} \end{equation} which is not positive definite anymore. Further, the eigenvalues of an anti-symmetric matrix is given by \begin{equation} F^{\mu}_{\ \ \ \ \nu} x^{\nu}=\lambda x^{\mu} \end{equation} where $x^{\mu}$ is the eigenvector.

Here is my question. If we are working in Euclidean space, the metric is $\{1,1,1,... \}$ and any real anti-symmetric matric has eigenvalues of the form $\{i \theta_1,-i \theta_1,i\theta_2,-i\theta_2,...\}$. I wonder if there is a similar theorem for the anti-symmetric matrix in Lorentz signature. For a very special $F^{\mu}_{\ \ \ \ \nu}$ \begin{equation}F^{\mu}_{\ \ \ \ \nu}=\left\{ \begin{array}{cccc} 0&a&0&0\\ a&0&0&0\\ 0&0&0&b\\ 0&0&-b&0 \end{array} \right\} \end{equation} the eigenvalues are $\{a,-a,ib,-ib\}$, so I guess the eigenvalues for arbitrary $F$should be $\{ \sigma,-\sigma,i \theta_1,-i \theta_1,i\theta_2,-i\theta_2,...\}$ where $\sigma$ and $\theta$ are all real, but I do not know whether that is true or how to prove that.

Thank you!

  • 1
    $\begingroup$ Use \langle and \rangle instead of < and > for inner products! It conforms to variables much better! $\endgroup$ – Cameron Williams Jun 15 '18 at 5:44

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