Every 2n-by-2n complex matrix is a sum of three symplectic matrices

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• 作者：Ralph John de la Cruz^a ; ^{rjdelacruz@math.upd.edu.ph} ; Dennis I. Merino^b ; ^{dmerino@selu.edu} ; Agnes T. Paras^a ; ^{agnes@math.upd.edu.ph}
• 关键词：15A21 ; 15A23
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：517
• 期：Complete
• 页码：199-206
• 全文大小：273 K
• 卷排序：517

文摘

Let J2n=[0In−In0]. An A∈M2n(C)A∈M2n(C) is called symplectic   if ATJ2nA=J2nATJ2nA=J2n. If n=1n=1, then we show that every matrix in M2n(C)M2n(C) is a sum of two symplectic matrices. If n>1n>1, then we show that every matrix in M2n(C)M2n(C) is a sum of three symplectic matrices; moreover, we show that some matrices cannot be written with less than three symplectic matrices. We also show that for every A∈M2n(C)A∈M2n(C), there exist symplectic P  , Q∈M2n(C)Q∈M2n(C) and B, C  , D∈Mn(C)D∈Mn(C) such that PAQ=[BC0D]. If A   is skew Hamiltonian (J2n−1ATJ2n=A), then we show that A is a sum of two symplectic matrices.