矩阵代数中可乘保范映照的显形式  

Explicit Forms of Norm Preserving Multiplicative Maps

作　　者：戴清平[1] 冯良贵[1] 

机构地区：[1]国防科技大学理学院,湖南长沙410073

出　　处：《国防科技大学学报》2008年第3期95-99,共5页Journal of National University of Defense Technology

基　　金：国家自然科学基金资助项目(10471045);新世纪优秀人才计划项目

摘　　要：在矩阵的可乘映照理论与应用中,判定一个可乘映照是否保持某类特定的数值特征以及获取可乘映照在保持某类数值特征条件下的解析式备受关注。对此,着重研究了一般可乘映照在具有某种保秩性或保范性下的表示问题。借助于构造的方法,给出了判定一个可乘映照是否为保秩映照的新的便捷方法。针对F=R或C,分别得到了Mn(F)上保1-范数、保∞-范数以及保F范数的可乘映照的显形式,进而证明了Mn(F)上保持1-范数可乘映照必为保F范数映照,而可乘保F范数映照又一定保谱半径、保数值半径、保正规性、保酉性等。With the multiplieafive map theory on matrix and its applications, a lot of attempts have been made for judging whether a multiplieative map can preserve certain desired numerical characters and obtain the explicit form of a multiplieafive map under the restriction of perserving some numerical characters. In this respect, multiplieatlve maps without assuming linearity on matrix algebra, which have certain rank preserving or norm preserving properties, are considered mainly in this paper. By virtue of a way of construetion, the complete descriptions of those maps are presented, and it is shown that a maximum column sum norm preserving multiplieative map is one of Frobenius norm and a Frobenius norm preserving multlplieative map must preserve spectral radius, numerical radius, normality, unitarity etc.. In particular, a new approach is also provided for judging whether a multiplicative map preserves rank.

关 键 词：可乘映照 矩阵范数 表示 

分 类 号：O174[理学—数学] O151.21[理学—基础数学]