  Consider the set of scalars $\alpha$ for which the $\alpha$th Hadamard power
of any $n\times n$ positive semi-definite (p.s.d.) matrix with non-negative
entries is p.s.d. It is known that this set is of the form $\{0, 1, \dots,
n-3\}\cup [n-2, \infty)$. A natural question is "what is the possible form of
the set of such $\alpha$ for a fixed p.s.d. matrix with non-negative entries?".
In all examples appearing in the literature, the set turns out to be union of a
finite set and a semi-infinite interval. In this article, examples of matrices
are given for which the set consists of a finite set and more than one disjoint
interval of positive length. In fact, it is proved that for some matrices, the
number of such disjoint intervals can be made arbitrarily large by taking $n$
large. The case when the entries of the matrices are not necessarily
non-negative is also considered.
