Weighted subdirect sum of matrices: Definition and properties for positivity classes

The concept of subdirect sum of matrices (a type of sum of matrices with overlapping blocks) was introduced in 1999 and has been broadly studied. In this paper we extend this concept by introducing the weighted subdirect sum: a sum of matrices with overlapping blocks allowing to give a different weight to the overlapped blocks. This concept naturally arises in applications with overlapping regions such as overlapping graphs in multilayer networks or in iterative methods to solve linear systems of equations based on overlapping blocks. We analyze the same positivity classes of matrices that were studied for the usual subdirect sum in the seminal paper of 1999. We extend those previous results on positivity classes to the new weighted subdirect sum, and we also introduce a methodology that can be useful to extend these results to other classes of matrices. We illustrate the theoretical results with some small examples and we show the potential applicability of the new concept in two lines of future research.