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The decomposition is $A=H+iS$. Both matrices are Hermitian: $H^{†}=H$ and $S^{†}=S$, directly from their definitions and complex conjugation of $1/(2i)$. The matrix $H$ is the Hermitian part of a matrix, whereas $S$ is the Hermitian matrix obtained by dividing the skew-Hermitian part by $i$.