Cauchy-Schwarz不等式的推广

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.

Extension of Cauchy-Schwarz Inequality

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.