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Additive Rank-1 Preservers Between Hermitian Matrix Spaces Over Quaternion Division Algebra

Han Jing-wen

Abstract

Let Q be the quaternion division algebra over real field F.Denote by H_n(Q)the set of all n×n hermitian matrices over Q.We characterize the additive maps from H_n(Q) into H_m(Q)that preserve rank-1 matrices when the rank of the image of I_n is equal to n. Let Q_R be the quaternion division algebra over the field of real number R.The additive maps from H_n(Q_R) into H_m(Q_R)that preserve rank-1 matrices are also given.

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Key concepts: Mathematics, Hermitian matrix, Quaternion, Quaternion algebra, Division algebra, Rank (graph theory), Division ring, Division (mathematics)