  We consider a sequence of matrices that are associated to Markov dynamical
systems and use determinant-free linear algebra techniques (as well as some
algebra and complex analysis) to rigorously estimate the eigenvalues of every
matrix simultaneously without doing any calculations on the matrices
themselves. As a corollary, we obtain mixing rates for every system at once, as
well as symmetry properties of densities associated to the system; we also find
the spectral properties of a sequence of related factor systems.
