  The CPA-laser is a coexisting state of coherent perfect absorption and lasing
that was proposed in parity-time ($\cal PT$) symmetric photonic systems. In
this work we show that the spectral signature of a CPA-laser displayed by the
singular value spectrum of the scattering matrix ($S$) can be orders of
magnitude wider than that displayed by the eigenvalue spectrum of $S$. Since
the former reflects how strongly light can be absorbed or amplified and the
latter announces the spontaneous symmetry breaking of $S$, these contrasting
spectral signatures indicate that near perfect absorption and extremely strong
amplification can be achieved even in the $\cal PT$-symmetric phase of $S$,
which is known for and defined by its flux-conserving eigenstates. We also show
that these contrasting spectral signatures are accompanied by strikingly
different sensitivities to disorder and imperfection, suggesting that the
eigenvalue spectrum is potentially suitable for sensing and the singular value
spectrum for robust switching.
