In this paper, we derive sufficient conditions on drift matrices under which block-diagonal solutions to Lyapunov inequalities exist. The motivation for the problem comes from a recently proposed basis pursuit algorithm. In particular, this algorithm can provide approximate solutions to optimisation programmes with constraints involving Lyapunov inequalities using linear or second order cone programming. This algorithm requires an initial feasible point, which we aim to provide in this paper. Our existence conditions are based on the so-called $mathcal{H}$ matrices. We also establish a link between $mathcal{H}$ matrices and an application of a small gain theorem to the drift matrix. We finally show how to construct these solutions in some cases without solving the full Lyapunov inequality.
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