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A matrix is called strictly sign-regular of order $k$ (denoted by $SSR_k$) ifall its $k\times k$ minors are non-zero and have the same sign. For example,totally positive matrices, i.e., matrices with all minors positive, are $SSR_k$for all $k$. Another important subclass are those that are $SSR_k$ for all odd$k$. Such matrices have interesting sign variation diminishing properties, andit has been recently shown that they play an important role in the analysis ofcertain nonlinear cooperative dynamical systems.  In this paper, the spectral properties of nonsingular matrices that are$SSR_k$ for a specific value $k$ are studied. One of the results is that theproduct of the first $k$ eigenvalues is real and of the same sign as the$k\times k$ minors, and that linear combinations of certain eigenvectors havespecific sign patterns. It is then shown how known results for matrices thatare $SSR_k$ for several values of $k$ can be derived from these spectralproperties.  Using these theoretical results, the notion of a totally positivediscrete-time system (TPDTS) is introduced. This may be regarded as thediscrete-time analogue of the important notion of a totally positivedifferential system, introduced by Schwarz in 1970. The asymptotic behavior oftime-invariant and time-varying TPDTSs is analyzed, and it is shown that everytrajectory of a periodic time-varying TPDTS converges to a periodic solution.

On the spectral properties of nonsingular matrices that are strictly  sign-regular for some order with applications to totally positive  discrete-time systems