  We characterize the limiting smallest eigenvalue distributions (or hard edge
laws) for sample covariance type matrices drawn from a spiked population. In
the case of a single spike, the results are valid in the context of the general
beta ensembles. For multiple spikes, the necessary construction restricts
matters to real, complex or quaternion (beta=1, 2, or 4) ensembles. The limit
laws are described in terms of a random integral operators, and partial
differential equations satisfied by the corresponding distribution functions
are derived as corollaries. We also show that, under a natural limit, all
spiked hard edge laws derived here degenerate to the critically spiked soft
edge laws (or deformed Tracy-Widom laws). The latter were first described at
beta=2 by Baik, Ben Arous, and Peche, and from a unified beta random operator
point of view by Bloemendal and Virag.
