Cauchy-Schwarz不等式的推广  

Extension of Cauchy-Schwarz Inequality

作　　者：李娟[1] 崔文泉[1] 

机构地区：[1]中国科学技术大学,合肥230026

出　　处：《大学数学》2006年第6期144-147,共4页College Mathematics

摘　　要：在非负定矩阵的偏序意义下讨论了对Cauchy-Schwarz不等式的推广,将随机变量情形下的Cauchy-Schwarz不等式推广到随机向量情形,而且两个随机向量的维数不要求相等,一个是随机变量另一个是随机向量是其中的一个特殊情形,另外还研究了有限维空间中的向量情形的Cauchy-Schwarz不等式在矩阵情形下的推广,得到一个十分简明的结果,并将此结果用于讨论一类随机向量簇的协方差阵的下界,不仅得到下界的具体表达式,而且给出能达到该下界的充分必要条件.Under the sense of partial ordering about non-negative definite matrices, we extend respectively the Cauchy-Schwarz inequality of two random variables to the case of two possible multi-dimensional random vectors as well as that of two finite dimensional vectors to the case of two matrices. From the extension result of multi-dimensional random vectors, we discuss and give the lower bound of covariance matrices of a given family of random vectors, and the sufficient and necessary condition to reach this lower bound is obtained.

关 键 词：CAUCHY-SCHWARZ不等式 偏序 随机向量 协方差阵 投影算子 

分 类 号：O211[理学—概率论与数理统计]