  We study factorizations of rational matrix functions with simple poles on the
Riemann sphere. For the quadratic case (two poles) we show, using
multiplicative representations of such matrix functions, that a good coordinate
system on this space is given by a mix of residue eigenvectors of the matrix
and its inverse. Our approach is motivated by the theory of discrete
isomonodromic transformations and their relationship with difference Painlev\'e
equations. In particular, in these coordinates, basic isomonodromic
transformations take the form of the discrete Euler-Lagrange equations.
Secondly we show that dPV equations, previously obtained in this context by D.
Arinkin and A. Borodin, can be understood as simple relationships between the
residues of such matrices and their inverses.
