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The Riemann -- Silberstein -- Majorana -- Oppenheimer approach to the Maxwellelectrodynamics in vacuum is investigated within the matrix formalism. Thematrix form of electrodynamics includes three real 4 \times 4 matrices. Withinthe squaring procedure we construct four formal solutions of the Maxwellequations on the base of scalar Klein -- Fock -- Gordon solutions. The problemof separating physical electromagnetic waves in the linear space\lambda_{0}\Psi^{0}+\lambda_{1}\Psi^{1}+\lambda_{2}\Psi^{2}+ lambda_{3}\Psi^{3}is investigated, several particular cases, plane waves and cylindrical waves,are considered in detail.

Maxwell equations in matrix form, squaring procedure, separating the variables, and structure of electromagnetic solutions