  A matrix is called strictly sign-regular of order $k$ (denoted by $SSR_k$) if
all 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, and
it has been recently shown that they play an important role in the analysis of
certain 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 the
product 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 have
specific sign patterns. It is then shown how known results for matrices that
are $SSR_k$ for several values of $k$ can be derived from these spectral
properties.
  Using these theoretical results, the notion of a totally positive
discrete-time system (TPDTS) is introduced. This may be regarded as the
discrete-time analogue of the important notion of a totally positive
differential system, introduced by Schwarz in 1970. The asymptotic behavior of
time-invariant and time-varying TPDTSs is analyzed, and it is shown that every
trajectory of a periodic time-varying TPDTS converges to a periodic solution.
