  Given $p$-dimensional Gaussian vectors $X_i \stackrel{iid}{\sim} N(0,
\Sigma)$, $1 \leq i \leq n$, where $p \geq n$, we are interested in testing a
null hypothesis where $\Sigma = I_p$ against an alternative hypothesis where
all eigenvalues of $\Sigma$ are $1$, except for $r$ of them are larger than $1$
(i.e., spiked eigenvalues).
  We consider a Rare/Weak setting where the spikes are sparse (i.e., $1 \ll r
\ll p$) and individually weak (i.e., each spiked eigenvalue is only slightly
larger than $1$), and discover a phase transition: the two-dimensional phase
space that calibrates the spike sparsity and strengths partitions into the
Region of Impossibility and the Region of Possibility. In Region of
Impossibility, all tests are (asymptotically) powerless in separating the
alternative from the null. In Region of Possibility, there are tests that have
(asymptotically) full power.
  We consider a CuSum test, a trace-based test, an eigenvalue-based Higher
Criticism test, and a Tracy-Widom test (Johnstone 2001), and show that the
first two tests have asymptotically full power in Region of Possibility.
  To use our results from a different angle, we derive new bounds for (a)
empirical eigenvalues, and (b) cumulative sums of the empirical eigenvalues,
both under the alternative hypothesis. Part (a) is related to those in Baik,
Ben-Arous and Peche (2005), but both the settings and results are different.
  The study requires careful analysis of the $L^1$-distance of our testing
problem and delicate Radom Matrix Theory. Our technical devises include (a) a
Gaussian proxy model, (b) Le Cam's comparison of experiments, and (c) large
deviation bounds on empirical eigenvalues.
