  The Riemann -- Silberstein -- Majorana -- Oppenheimer approach to the Maxwell
electrodynamics in presence of electrical sources and arbitrary media is
investigated within the matrix formalism. The symmetry of the matrix Maxwell
equation under transformations of the complex rotation group SO(3.C) is
demonstrated explicitly. In vacuum case, the matrix form includes four real $4
\times 4$ matrices $\alpha^{b}$. In presence of media matrix form requires two
sets of $4 \times 4$ matrices, $\alpha^{b}$ and $\beta^{b}$ -- simple and
symmetrical realization of which is given. Relation of $\alpha^{b}$ and
$\beta^{b}$ to the Dirac matrices in spinor basis is found. Minkowski
constitutive relations in case of any linear media are given in a short
algebraic form based on the use of complex 3-vector fields and complex
orthogonal rotations from SO(3.C) group. The matrix complex formulation in the
Esposito's form,based on the use of two electromagnetic 4-vectors,
$e^{\alpha}(x) = u_{\beta} F^{\alpha \beta}(x), b^{\alpha} (x) = u_{\beta}
\tilde{F}^{\alpha \beta}(x) $ is studied and discussed. It is argued that
Esposito form is achieved trough the use of a trivial identity
$I=U^{-1}(u)U(u)$ in the Maxwell equation.
