  When we speak about parametric programming, sensitivity analysis, or related
topics, we usually mean the problem of studying specified perturbations of the
data such that for a given optimization problem some optimality criterion
remains satisfied. In this paper, we turn to another question. Suppose that $A$
is a matrix having a specific property $\mathcal{P}$. What are the maximal
allowable variations of the data such that the property still remains valid for
the matrix? We study two basic forms of perturbations. The first is a
perturbation in a given direction, which is closely related to parametric
programming. The second type consists of all possible data variations in a
neighbourhood specified by a certain matrix norm; this is related to the
tolerance approach to sensitivity analysis, or to stability. The matrix
properties discussed in this paper are positive definiteness; P-matrix,
H-matrix and P-matrix property; total positivity; inverse M-matrix property and
inverse nonnegativity.
