There is an extensive lore regarding the properties of positive matrices, dating from the work of Frobenius and Perron at the beginning of the century, and ably summarized in several contemporary textbooks, such as Matrix Iterative Analysis by Varga [18], Non-Negative Matrices by Seneta [19], and Nonnegative Matrices in the Mathematical Sciences by Berman and Plemmons [20]. A recent addition to this series is Minc's Nonnegative Matrices.
Applied to topological matrices, the theory yields estimates for such things as the increase in the numbers of paths through a network as the path length increases. Applied to probabilistic matrices it allows the determination of equilibrium configurations and the rapidity of approach to equilibrium from arbitrary initial configurations.
It is not surprising that the theory of positive matrices has strong applications to Markov chains and similar probabilistic concepts, or even that much of the motivation for their study came from this direction.