In the analysis of a recently proposed distributed estimation algorithm based on the Kalman filtering and on gossip iterations, we needed to apply a new inequality which is valid for i.id. matrix valued random processes. This inequality can be useful in the analysis of the convergence rate of general jump Markov linear systems. In this paper, we present this inequality. This is based on the theory of majorization and on its use in the analysis of the singular values. Finally we will show the impact of this inequality on the performance analysis of gossip based distributed Kalman filters. (C) 2011 Elsevier Ltd. All rights reserved.