**Manuel De la Sen**

Institute of Research and Development of Processes IIDP, University of the Basque Country, Campus of Leioa, Leioa, PO Box 48940, Bizkaia, Spain; manuel.delasen@ehu.eus

Received: 8 May 2019; Accepted: 21 May 2019; Published: 24 May 2019

**Abstract:** This paper links the celebrated Cauchy's interlacing theorem of eigenvalues for partitioned updated sequences of Hermitian matrices with stability and convergence problems and results of related sequences of matrices. The results are also applied to sequences of factorizations of semidefinite matrices with their complex conjugates ones to obtain sufficiency-type stability results for the factors in those factorizations. Some extensions are given for parallel characterizations of convergen<sup>t</sup> sequences of matrices. In both cases, the updated information has a Hermitian structure, in particular, a symmetric structure occurs if the involved vector and matrices are complex. These results rely on the relation of stable matrices and convergen<sup>t</sup> matrices (those ones being intuitively stable in a discrete context). An epidemic model involving a clustering structure is discussed in light of the given results. Finally, an application is given for a discrete-time aggregation dynamic system where an aggregated subsystem is incorporated into the whole system at each iteration step. The whole aggregation system and the sequence of aggregated subsystems are assumed to be controlled via linear-output feedback. The characterization of the aggregation dynamic system linked to the updating dynamics through the iteration procedure implies that such a system is, generally, time-varying.

**Keywords:** Aggregation dynamic system; Discrete system; Epidemic model; Cauchy's interlacing theorem; Output-feedback control; Stability; Antistable/Stable matrix
