  The exact parameter values of mathematical models are often uncertain or even
unknown. Nevertheless, we may have access to crude information about the
parameters, e.g., that some of them are nonzero. Such information can be
captured by so-called pattern matrices, whose symbolic entries are used to
represent the available information about the corresponding parameters. In this
paper, we focus on pattern matrices with three types of symbolic entries: those
that represent zero, nonzero, and arbitrary parameters. We formally define and
study addition and multiplication of such pattern matrices. The results are
then used in the study of three strong structural properties, namely,
controllability of linear descriptor systems, and input-state observability and
output controllability of linear systems.
