In Riemannian geometry we require that the metric $g$ be a positive definite real matrix and in semi-Riemannian geometry we relax this requirement and instead ask that the metric be a non degenerate symmetric matrix. If we relax further and simply ask that the matrix be non degenerate, that is, we allow for asymmetric matrices with non zero determinant, what kind of object do we get? Does this object have a name?
Similarly, if we generalize further and allow the matrix to be any complex matrix with non zero determinant, does this have a name?
Finally, what if we allow the matrix to be complex valued but require it to be self adjoint, what is this called?
I would like to know if any of these objects have been studied and if so what they are called. Any references are greatly appreciated. If you need clarification please let me know.