Representation of doubly infinite matrices as non-commutative Laurent series

We present a new way to deal with doubly infinite lower Hessenberg matrices based on the representation of the matrices as the sum of their diagonal submatrices. We show that such representation is a simple and useful tool for computation purposes and also to obtain general properties of the matrices related with inversion, similarity, commutativity, and Pincherle derivatives. The diagonal representation allows us to consider the ring of doubly infinite lower Hessenberg matrices over a ring R as a ring of Laurent series in one indeterminate, with coefficients in the ring of R-valued sequences that don’t commute with the indeterminate