  In this paper, we describe sufficient conditions when block-diagonal
solutions to Lyapunov and $\mathcal{H}_{\infty}$ Riccati inequalities exist. In
order to derive our results, we define a new type of comparison systems, which
are positive and are computed using the state-space matrices of the original
(possibly nonpositive) systems. Computing the comparison system involves only
the calculation of $\mathcal{H}_{\infty}$ norms of its subsystems. We show that
the stability of this comparison system implies the existence of block-diagonal
solutions to Lyapunov and Riccati inequalities. Furthermore, our proof is
constructive and the overall framework allows the computation of block-diagonal
solutions to these matrix inequalities with linear algebra and linear
programming. Numerical examples illustrate our theoretical results.
