设Q表示四元数集合, Mn ( Q)表示n ×n四元数矩阵的集合。若M、N∈Mn ( Q)分别是下三角可逆四元数矩阵且φ( A)=MAN,证明了对于任意下三角四元数矩阵A∈Mn ( Q),如果φ( A)与A具有相同的左特征值,当且仅当M、N和A中的元素mss,nss和ass的虚部对应成比例,且ms nss =1,或虚部对应为零。%Let Q be the set of quaternion numbers and Mn(Q) the set of n ×n quaternion matrices.If M and N∈Mn ( Q) are lower triangular quaternion matrices and the linear mapφ:Mn ( Q)→Mn ( Q) satisfies that φ( A)=MAN, in this paper , we prove that for arbitrary lower triangular quaternion matrices A if the linear map φsatisfies that φ(A) is of the same left eigenvalue as that of A if and only if the imagine parts of elements mss∈M,nss∈N and ass∈A are corresponding proportion and mssnss=1.
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