# Eigenvalues of an anti-symmetric matrix in Lorentz signature

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 $$\langle u,v \rangle=u^{\mu}v^{\nu}\eta_{\mu \nu}$$ which is not positive definite anymore. Further, the eigenvalues of an anti-symmetric matrix is given by $$F^{\mu}_{\ \ \ \ \nu} x^{\nu}=\lambda x^{\mu}$$ 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 $F^{\mu}_{\ \ \ \ \nu}$ $$F^{\mu}_{\ \ \ \ \nu}=\left\{ \begin{array}{cccc} 0&a&0&0\\ a&0&0&0\\ 0&0&0&b\\ 0&0&-b&0 \end{array} \right\}$$ 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!

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