Matrix Wielandt inequality via the matrix geometric mean

In this paper, by virtue of the matrix geometric mean and the polar decomposition, we present new Wielandt type inequalities for matrices of any size. To this end, based on results due to J.I. Fujii, we reform a matrix Cauchy-Schwarz inequality, which differs from ones due to Marshall and Olkin. As an application, we show a new block matrix version of Wielandt type inequalities under the block rank additivity condition.