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This paper is concerned with the dual-LMI-based analysis of linear positive systems. As the first contribution, we will show that the cerebrated Perron-Frobenius theorem can be proved concisely via a duality-based argument. On the other hand, in the second part of the paper, we extend the well-known result that a stable Metzler matrix admits a diagonal Lyapunov matrix as the solution of the Lyapunov inequality. More precisely, again via a duality-based argument, we will clarify a necessary and sufficient condition under which a stable Metzler matrix admits a diagonal Lyapunov matrix with some identical diagonal entries. This new result leads to an alternative proof for the recent result by Tanaka and Langbort on the existence of a diagonal Lyapunov matrix for the LMI characterizing the L2 gain of positive systems.