  This paper investigates the signal detection problem in colored noise with an
unknown covariance matrix. In particular, we focus on detecting an unknown
non-random signal by capitalizing on the leading eigenvalue of the whitened
sample covariance matrix as the test statistic (a.k.a. Roy's largest root
test). Since the unknown signal is non-random, the whitened sample covariance
matrix turns out to have a non-central $F$-distribution. This distribution
assumes a singular or non-singular form depending on whether the number of
observations $p\lessgtr$ the system dimensionality $m$. Therefore, we
statistically characterize the leading eigenvalue of the singular and
non-singular $F$-matrices by deriving their cumulative distribution functions
(c.d.f.). Subsequently, they have been utilized in deriving the corresponding
receiver operating characteristic (ROC) profiles. We also extend our analysis
into the high dimensional domain. It turns out that, when the signal is
sufficiently strong, the maximum eigenvalue can reliably detect it in this
regime. Nevertheless, weak signals cannot be detected in the high dimensional
regime with the leading eigenvalue.
