After the preparations of the previous chapter, we are now in a position to introduce our notion of connection matrices in a purely algebraic way. While the definition is modeled on previous work by Robbin and Salamon, as well as Harker, Mischaikow, and Spendlove, their approach has to be extended to allow for varying underlying posets. For this, we first introduce the notion of reduced filtered chain complexes, before discussing Conley complexes and connection matrices, as well as their existence for arbitrary poset filtered chain complexes. We close the chapter with a new equivalence relation for Conley complexes, which enables us to precisely formulate the uniqueness question of connection matrices for the first time. Despite being a completely algebraic criterion based on the notion of essentially graded morphisms, it will later allow us to detect underlying bifurcations in combinatorial dynamics.