  Positive definite (p.d.) matrices arise naturally in many areas within
mathematics and also feature extensively in scientific applications. In modern
high-dimensional applications, a common approach to finding sparse positive
definite matrices is to threshold their small off-diagonal elements. This
thresholding, sometimes referred to as hard-thresholding, sets small elements
to zero. Thresholding has the attractive property that the resulting matrices
are sparse, and are thus easier to interpret and work with. In many
applications, it is often required, and thus implicitly assumed, that
thresholded matrices retain positive definiteness. In this paper we formally
investigate the algebraic properties of p.d. matrices which are thresholded. We
demonstrate that for positive definiteness to be preserved, the pattern of
elements to be set to zero has to necessarily correspond to a graph which is a
union of disconnected complete components. This result rigorously demonstrates
that, except in special cases, positive definiteness can be easily lost. We
then proceed to demonstrate that the class of diagonally dominant matrices is
not maximal in terms of retaining positive definiteness when thresholded.
Consequently, we derive characterizations of matrices which retain positive
definiteness when thresholded with respect to important classes of graphs. In
particular, we demonstrate that retaining positive definiteness upon
thresholding is governed by complex algebraic conditions.
