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Suppose that $H$ is orthogonally additive and orthogonally multiplicative on self-adjoint elements. We show that either the range of $H$ consists of zero trace elements, or there is a scalar sequence $\\{\\lambda_n\\}$ and an invertible $S$ in $M_m$ such that $$ H(x) =\\sum_{n\\geq 1} \\lambda_n S^{-1}x^nS, \\quad\\forall x \\in M_m,%\\eqno{(\\ddag)} $$ or $$ H(x) =\\sum_{n\\geq 1} \\lambda_n S^{-1}(x^t)^nS, \\quad\\forall x \\in M_m. $$ Here, $x^t$ is the transpose of the matrix $x$. 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