  This paper is motivated by computing correlations for domino tilings of the
Aztec diamond. It is inspired by two of the three distinct methods that have
recently been used in the simplest case of a doubly periodic weighting, that is
the two-periodic Aztec diamond. One of the methods, powered by the domino
shuffle, involves inverting the Kasteleyn matrix giving correlations through
the local statistics formula. Another of the methods, driven by a Wiener-Hopf
factorization for two-by-two matrix valued functions, involves the Eynard-Mehta
theorem. For arbitrary weights the Wiener-Hopf factorization can be replaced by
an LU- and UL-decomposition, based on a matrix refactorization, for the product
of the transition matrices. This paper shows that, for arbitrary weightings of
the Aztec diamond, the evolution of the face weights under the domino shuffle
and the matrix refactorization is the same. In particular, these dynamics can
be used to find the inverse of the LGV matrix in the Eynard-Mehta Theorem.
