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

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