Inverse this Stokes Matrix.
In fact, $M$ has no inverse. To see that this is the case, note that the vector $\xi$ satisfies $$ M_{\xi}\xi = (\xi \otimes \xi - |\xi^2|I)\xi = (\xi \otimes \xi)\xi - |\xi^2|\xi = |\xi|^2\xi - |\xi|^2\xi = 0. $$ Because $M$ has a non-trivial nullspace, it cannot be invertible.