Matrix Versions 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 Lo|¨wner 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, what happens to the inequality when the positive definite matrix is allowed to be positive semi-definite was considered. This inequality is a generalization of the inequality Wang and Ip gave.