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

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