交错阵的保秩等价的加法映射

【作者】 陈雪梅；

【导师】 唐孝敏；

【作者基本信息】 黑龙江大学 ， 基础数学， 2007， 硕士

【摘要】 线性保持问题是矩阵论中一个热门的研究领域，主要刻划矩阵空间的保不变量(函数，子集，关系)的线性算子和加法算子。对于矩阵空间上保持某些等价关系的研究已取得了一些成果，例如，Li等研究了在双射的条件下，复数域上全矩阵空间的保持秩等价的非零线性算子；Horn等研究了复数域上全矩阵空间的保持秩等价的非零线性算子。许多研究者对不同矩阵集上的加法或线性保持问题感兴趣。交错阵与二次型和典型群中的辛群有密切的关系，同时，交错阵集合也构成线性李代数中的正交李代数。因此，研究交错阵的保持问题是很有价值和非常有趣的。设F是一个任意域，F~*是F中所有非零元素，m，n≥4是任意的正整数。对F上的任意长方阵A，以A~t和rankA分别表示其转置及秩。如果矩阵A满足A~t=-A且A的对角线上元素全为0，则称A为交错矩阵。令K_n(F)是F上n×n交错阵空间。最近，唐孝敏，陈雪梅等研究了交错阵空间上保秩等价的非零线性算子，本文主要把上述文献的结果由线性推广到加法，应用交错阵矩阵几何基本定理刻划出从K_n(F)到K_n(F)的保秩等价的加法映射，并给出了以下几个方面的应用：(1)刻划了从K_n(F)到K_n(F)的保秩的加法映射，推出了保秩的加法映射一定是保秩等价的加法映射，从而把保秩等价的结果应用到保秩上。(2)证明了φ保持秩不增关系，保秩等价和保秩三者是等价的，从而应用保秩等价的结果刻划出保持上述秩关系的加法映射。更多还原

【Abstract】 Linear preserver problems are an active research area in matrix theory. Itmainly characterizes the linear and additive operators preserving invariant（funtion,subset, relation）. There are some results preserving certain equivalence relationson matrix spaces. For example, Li e.t.al, studied nonzero linear operator that pre-serves rank equivalence under the condition of bijectivity on M_m×n（C）, where Cis the field of complex numbers, Horn e.t.al, studied nonzero linear operator thatpreserves rank equivalence without the condition of bijectivity on M_m×n（C）. It isinteresting for many researchers to study the linear and additive preserver problemson different matrices. Alternate matrices have close relation with quadratic formsand symplectic group of classical groups, meanwhile, orthogonal Lie algebra of lin-ear Lie algebra consists of alternate matrices. So it is worthwhile and interestingto study the preserving problems of alternate matrices. Let F be any field, F^* beits subset consisting of all nonzero elements, and m, n≥4 be any integer. ForA∈K_n（F）, we denote by A^t the transpose of A, denote by rank（A） the rank of A.A square matrix A is said to be alternate if A^t=-A and all diagonal elements arezeros. Denote by K_n（F） the space of all n×n alternate matrices over F. Recently,Tang Xiaomin, Chen Xuemei, studied nonzero linear operator that preserved rankequivalence from K_n（F） to K_n（F）. In this paper, we extend the above referenceresult to addition, we use alternate matrix geometric fundamental theorem to char-acterize the additive operator preserving rank equivalence from K_n（F） to K_n（F）,some applications include severals aspects as following:（1） characterizing the additive operator preserving rank from K_n（F） to K_n（F）,obtaining that the additive operator preserving rank is necessarily preserving rankequivalence and applying the result preserving rank equivalence to preserving rank.（2） proving that preserving rank non-increasing relation, preserving rank equiv-alence, and preserving rank are all equivalent. Furthermore, we characterize the ad-ditive operator preserving the above rank relation using the result preserving rankequivalence.更多还原