推广的Wielandt不等式的矩阵形式

Wielandt不等式是对著名的Cauchy-Schwarz不等式的一种改进，1999年，王松桂等人把Wielandt不等式推广到x和y为矩阵的情形，并给出了许多统计应用.本文依照张宝学等人的研究方法，在Loewner偏序关系下，对于半正定Hermite阵，给出了推广的Wielandt不等式的矩阵形式，从而进一步推广了王松桂等人的对于正定Hermite阵，给出的Wielandt不等式的矩阵形式的结果，Wielandt不等式的矩阵形式被推广到奇异的情形．结果表明，我们得到的不等式是王松桂等的结果的更一般形式的表达式．

A Generalized Matrix Version of the Wielandt Inequality

The Wielandt inequality is an improvement on the general Cauchy-Schwarz inequality,and its applications to statistics were studied.In a note,Wang and Ip(1999) gave the Wielandt inequality in matrix version in terms of the Loewner partial ordering.That inequality was an extension of the well-known Wielandt inequality in which both x and y are vectors.Some applications to statistics were also given.In this paper,motivated by a note of Zhang Bao Xue and Zhu Xian Hai,what happens to the inequality when the positive definite matrix is allowed to be positive semidefinite was considered.The matrix version of the Wielandt inequality is extended to the singular case.This inequality is a generalization of the inequality Wang and Ip gave.