2018/10/26 by Rola Alseidi, Alseidi, Rola, Michael Margaliot +3
Computer Science · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Matrix Theory and Algorithms #Quantum chaos and dynamical systems #Quantum optics and atomic interactions
paper · pdf · doi:10.48550/arxiv.1810.11358
openalex publication_date 2018/10/26 · openalex created_date 2022/09/29 · openalex updated_date 2026/07/28
A matrix is called strictly sign-regular of order k (denoted by SSRk) if\nall its k\× k minors are non-zero and have the same sign. For example,\ntotally positive matrices, i.e., matrices with all minors positive, are SSRk\nfor all k. Another important subclass are those that are SSRk for all odd\nk. Such matrices have interesting sign variation diminishing properties, and\nit has been recently shown that they play an important role in the analysis of\ncertain nonlinear cooperative dynamical systems.\n In this paper, the spectral properties of nonsingular matrices that are\nSSRk for a specific value k are studied. One of the results is that the\nproduct of the first k eigenvalues is real and of the same sign as the\nk\× k minors, and that linear combinations of certain eigenvectors have\nspecific sign patterns. It is then shown how known results for matrices that\nare SSRk for several values of k can be derived from these spectral\nproperties.\n Using these theoretical results, the notion of a totally positive\ndiscrete-time system (TPDTS) is introduced. This may be regarded as the\ndiscrete-time analogue of the important notion of a totally positive\ndifferential system, introduced by Schwarz in 1970. The asymptotic behavior of\ntime-invariant and time-varying TPDTSs is analyzed, and it is shown that every\ntrajectory of a periodic time-varying TPDTS converges to a periodic solution.\n