I've been looking for a set of 4 matrices:
$$A^\mu A^\nu = \eta^{\mu\nu}$$
I'm expecting maybe 4x4 complex-valued matrices, I've been trying everything by combining Dirac's gamma matrices but I can't quite get it to work out.
If you happen to know that this is only possible for higher NxN I would love to know!
I'm interested in any kind of set that is similar, perhaps involving some hermitian conjugates?
This is not possible. $\eta^{\mu\nu}$ is a scalar and I assume each $A^\mu$ is a matrix. You cannot multiply two matrices of the same size like $A^\mu A^\mu$ and get a scalar (unless they are 1${\times}$1).
Even if we interpret $\eta^{\mu\nu}$ as $\eta^{\mu\nu}I$ with $I$ the identity matrix this is still impossible since $$ A^\mu A^\mu = \eta^{\mu\mu}I \ne 0 $$ means each $A^\mu$ is invertible, but $$ A^\mu A^\nu = \eta^{\mu\nu}I = 0 $$ for $\mu\ne\nu$ means $A^\mu$ is not invertible. This is a contradiction.