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Diagonally dominant matrices have many applications in systems and controltheory. Linear dynamical systems with scaled diagonally dominant driftmatrices, which include stable positive systems, allow for scalable stabilityanalysis. For example, it is known that Lyapunov inequalities for this class ofsystems admit diagonal solutions. In this paper, we present an extension ofscaled diagonally dominance to block partitioned matrices. We show that ourdefinition describes matrices admitting block-diagonal solutions to Lyapunovinequalities and that these solutions can be computed using linear algebraictools. We also show how in some cases the Lyapunov inequalities can bedecoupled into a set of lower dimensional linear matrix inequalities, thusleading to improved scalability. We conclude by illustrating some advantagesand limitations of our results with numerical examples.

Block-Diagonal Solutions to Lyapunov Inequalities and Generalisations of Diagonal Dominance