矩阵空间之间的保持问题
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摘要
刻画矩阵集之间保持不变量的映射结构问题被称为保持问题.近几十年来,保持问题已成为国际矩阵论研究中一个十分活跃的领域.这一方面是因为它具有重要的理论价值;另一方面是因为许多问题在量子力学、微分几何、微分方程、系统控制和数理统计等领域有着广泛的实际应用背景;再者,通过对保持问题的研究可以得到关于矩阵的不变量、函数、集合和关系等重要理论成果.
从映射的角度来说,保持问题可分为:线性保持问题、加法保持问题和更一般的保持问题.从保持的不变量的角度来说,保持问题可分为:保持子集、保持关系、保持函数和保持变换.
本文针对矩阵空间之间的几个保持问题进行了系统的研究,概括起来有以下几个方面:
(1)利用交错矩阵空间K_n(F)上保持秩2和秩4矩阵的结论,刻画了不同维的交错矩阵空间之间保持伴随矩阵的线性映射φ: K_n(F)→K__m(F)的形式,证明其可以归结到同维的情形.
(2)利用上三角矩阵空间Tn(F)上保持秩1矩阵的的结论,刻画了Tn(F)上保持秩可加的线性映射的形式.同时,作为应用,还刻画了Tn(F)上保持秩可减的线性映射的形式,以及Tn(F)上使得“rank(A + B) = |rankA - rankB| (?)rankφ(A + B) = |rankφ(A) - rankφ(B)|”成立的线性映射φ的形式.
(3)就域F的特征不为2和为2两种情况,分别刻画了从S_n(F)到M_m(F)及从Sn(F)到S_m(F)保持群逆的线性映射的形式.
(4)利用上三角矩阵空间T_n(F)上保持秩1矩阵的结论,刻画了T_n(F)上保持秩交换的加法满射的形式.
(5)刻画了特征不为2的体K上矩阵空间Mn(K)上保持某种非平凡乘性矩阵函数的加法满射的形式.同时,作为应用,分别刻画了特征不为2的体K上矩阵空间M_n(K)上的保持Dieudonné行列式和保持可逆矩阵的加法满射,以及保持秩可加的加法双射,同时也刻画了四元数体Q上矩阵空间M_n(Q)上保持行列式detq的加法满射.
Preserving problems concern the characterization of maps between matrixspaces that preserve some invariants. During the past few decades, one of the veryactive research areas in matrix theory is the study of preserving problems. Firstly, ithas important theoretical values. Secondly, it has wide applications in other areas,such as, quantum mechanics, differential geometry, differential equation, system con-trol, mathematical statistics, and so on. Thirdly, studying preserving problems canhelp us to understand matrix invariants, functions, sets and relations.
According to the property of maps, preserving problems can be divided intothree categories, that is, linear preserving problems, additive preserving problemsand general preserving problems. According to the property of invariants, preservingproblems can be divided into four categories, that is, preserving subsets, preservingrelations, preserving functions and preserving transformations.
This paper considers some preserving problems on matrix spaces and obtains thefollowing five results:
(1) By using the conclusion of linear maps on alternate matrix space which pre-serve rank 2 and rank 4, the structure of linear mapφ: K_n(F)→K_m(F) whichpreserves adjoint matrices is characterized. It can be concluded that the characteriza-tion of alternate matrix spaces of different dimensions can be induced to characterizethe same dimensions.
(2) The characterization of linear mapφ: Tn(F)→Tn(F) which pre-serves rank-additivity is given. And then, applications to several related preserv-ing problems are considered. The linear mapsφ: T_n(F)→T_n(F) which pre-serve rank-subtractivity, or satisfy rank(A + B) = |rankA - rankB| which impliesrankφ(A + B) = |rankφ(A) - rankφ(B)|, are characterized respectively.
(3) The linear mapsφ: S_n(F)→M_m(F) andφ: S_n(F)→S_m(F) whichpreserve group inverses, are characterized respectively, where the character of thefield F is distinguished into two cases, that is chF = 2 and chF = 2.
(4) The structure of additive surjective mapφ: T_n(F)→T_n(F) which preservesrank commutativity is obtained.
(5) An additive surjective mapφ: M_n(K)→M_n(K) which preserves anon-trivial multiplicative matrix function is characterized, where K is a divisionring whose character is not 2. As applications, some additive surjective mapsφ: M_n(K)→M_n(K) which preserve the Dieudonn′e determinant, invertible ma-trices, are characterized respectively. Some additive bijective maps which preserverank-additivity are also characterized. And then, the structure of additive surjectivemapφ: M_n(Q)→M_n(Q) which preserves detq is obtained, where Q is a quaternionfield.
从映射的角度来说,保持问题可分为:线性保持问题、加法保持问题和更一般的保持问题.从保持的不变量的角度来说,保持问题可分为:保持子集、保持关系、保持函数和保持变换.
本文针对矩阵空间之间的几个保持问题进行了系统的研究,概括起来有以下几个方面:
(1)利用交错矩阵空间K_n(F)上保持秩2和秩4矩阵的结论,刻画了不同维的交错矩阵空间之间保持伴随矩阵的线性映射φ: K_n(F)→K__m(F)的形式,证明其可以归结到同维的情形.
(2)利用上三角矩阵空间Tn(F)上保持秩1矩阵的的结论,刻画了Tn(F)上保持秩可加的线性映射的形式.同时,作为应用,还刻画了Tn(F)上保持秩可减的线性映射的形式,以及Tn(F)上使得“rank(A + B) = |rankA - rankB| (?)rankφ(A + B) = |rankφ(A) - rankφ(B)|”成立的线性映射φ的形式.
(3)就域F的特征不为2和为2两种情况,分别刻画了从S_n(F)到M_m(F)及从Sn(F)到S_m(F)保持群逆的线性映射的形式.
(4)利用上三角矩阵空间T_n(F)上保持秩1矩阵的结论,刻画了T_n(F)上保持秩交换的加法满射的形式.
(5)刻画了特征不为2的体K上矩阵空间Mn(K)上保持某种非平凡乘性矩阵函数的加法满射的形式.同时,作为应用,分别刻画了特征不为2的体K上矩阵空间M_n(K)上的保持Dieudonné行列式和保持可逆矩阵的加法满射,以及保持秩可加的加法双射,同时也刻画了四元数体Q上矩阵空间M_n(Q)上保持行列式detq的加法满射.
Preserving problems concern the characterization of maps between matrixspaces that preserve some invariants. During the past few decades, one of the veryactive research areas in matrix theory is the study of preserving problems. Firstly, ithas important theoretical values. Secondly, it has wide applications in other areas,such as, quantum mechanics, differential geometry, differential equation, system con-trol, mathematical statistics, and so on. Thirdly, studying preserving problems canhelp us to understand matrix invariants, functions, sets and relations.
According to the property of maps, preserving problems can be divided intothree categories, that is, linear preserving problems, additive preserving problemsand general preserving problems. According to the property of invariants, preservingproblems can be divided into four categories, that is, preserving subsets, preservingrelations, preserving functions and preserving transformations.
This paper considers some preserving problems on matrix spaces and obtains thefollowing five results:
(1) By using the conclusion of linear maps on alternate matrix space which pre-serve rank 2 and rank 4, the structure of linear mapφ: K_n(F)→K_m(F) whichpreserves adjoint matrices is characterized. It can be concluded that the characteriza-tion of alternate matrix spaces of different dimensions can be induced to characterizethe same dimensions.
(2) The characterization of linear mapφ: Tn(F)→Tn(F) which pre-serves rank-additivity is given. And then, applications to several related preserv-ing problems are considered. The linear mapsφ: T_n(F)→T_n(F) which pre-serve rank-subtractivity, or satisfy rank(A + B) = |rankA - rankB| which impliesrankφ(A + B) = |rankφ(A) - rankφ(B)|, are characterized respectively.
(3) The linear mapsφ: S_n(F)→M_m(F) andφ: S_n(F)→S_m(F) whichpreserve group inverses, are characterized respectively, where the character of thefield F is distinguished into two cases, that is chF = 2 and chF = 2.
(4) The structure of additive surjective mapφ: T_n(F)→T_n(F) which preservesrank commutativity is obtained.
(5) An additive surjective mapφ: M_n(K)→M_n(K) which preserves anon-trivial multiplicative matrix function is characterized, where K is a divisionring whose character is not 2. As applications, some additive surjective mapsφ: M_n(K)→M_n(K) which preserve the Dieudonn′e determinant, invertible ma-trices, are characterized respectively. Some additive bijective maps which preserverank-additivity are also characterized. And then, the structure of additive surjectivemapφ: M_n(Q)→M_n(Q) which preserves detq is obtained, where Q is a quaternionfield.
引文
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