Relative entropy and discrete Poincaré inequalities for reducible matrices

A general relative entropy functional has been used recently in Perthame (2007) [3] to provide a uniform treatment of various estimates of the decay of the exponential function ( e t A ) t ≥ 0 , where A is a matrix with positive off-diagonal entries. In this note we show that the method can be extended to general irreducible matrices. For reducible matrices, on the other hand, we show that staying within the framework of Perthame (2007) [3] only allows for control of the evolution in certain invariant subspaces of A .