In this dissertation, we begin by studying nonnegative matrices under the minus partial order. The study of the minus partial order was initiated by Hartwig and Nambooripad independently. We first give the precise structure of a nonnegative matrix dominated by a group-monotone matrix under this partial order. The special cases of stochastic and doubly stochastic matrices are also considered. We also derive the class of nonnegative matrices dominated by a nonnegative idempotent matrix. This improves upon and generalizes previously known results of Bapat, Jain and Snyder.
Next, we introduce the direct sum partial order and investigate the relationship among different partial orders, namely, the minus, direct sum, and Loewner partial orders on a von Neumann regular ring. It is proven that the minus and direct sum partial orders are equivalent on a von Neumann regular ring. On the set of positive semidefinite matrices, we show that the direct sum partial order implies the Loewner partial order. Various properties of the direct sum and minus partial orders are presented. We provide answers to two of Hartwig's questions regarding the minus partial order. One of the main results gives an explicit form of maximal
elements in a given subring. Our result generalizes the concept of a shorted operator of electrical circuits, as given by Anderson-Trapp. As an application of the main theorem, the unique shorted operator has been derived.
Finally, we consider the parallel sum of two matrices over a regular ring. Previously known results of Mitra-Odell and Hartwig are generalized. We also obtain a result on the harmonic mean of two idempotents.