On dual and minimal phase-type representations

In this paper the authors present some properties of the dual representation for a given representation of a phase-type distribution. The dual representation is a representation of the original distribution which is the time reversal of the initial one. As an application of this notion they give a characterisation of the representations whose dimension is equal to the algebraic degree of the distribution. This characterisation is expressed in terms of the recently introduced notion of simplicitly. It turns out that the property holds true when the original distribution and the dual one are simple.