  Thanks to its versatility, its simplicity, and its fast convergence, ADMM is
among the most widely used approaches for solving a convex problem in
distributed form. However, making it running efficiently is an art that
requires a fine tuning of system parameters according to the specific
application scenario, and which ultimately calls for a thorough understanding
of the hidden mechanisms that control the convergence behaviour. In this
framework we aim at providing new theoretical insights on the convergence
process and specifically on some constituent matrices of ADMM whose
eigenstructure provides a close link with the algorithm's convergence speed.
One of the key technique that we develop allows to effectively locate the
eigenvalues of a (symmetric) matrix product, thus being able to estimate the
contraction properties of ADMM. In the comparison with the results available
from the literature, we are able to strengthen the precision of our speed
estimate thanks to the fact that we are solving a joint problem (i.e., we are
identifying the spectral radius of the product of two matrices) in place of two
separate problems (the product of two matrix norms).
