  We provide a general formula for the eigenvalue density of large random
$N\times N$ matrices of the form $A = M + LJR$, where $M$, $L$ and $R$ are
arbitrary deterministic matrices and $J$ is a random matrix of zero-mean
independent and identically distributed elements. For $A$ nonnormal, the
eigenvalues do not suffice to specify the dynamics induced by $A$, so we also
provide general formulae for the transient evolution of the magnitude of
activity and frequency power spectrum in an $N$-dimensional linear dynamical
system with a coupling matrix given by $A$. These quantities can also be
thought of as characterizing the stability and the magnitude of the linear
response of a nonlinear network to small perturbations about a fixed point. We
derive these formulae and work them out analytically for some examples of $M$,
$L$ and $R$ motivated by neurobiological models. We also argue that the
persistence as $N\rightarrow\infty$ of a finite number of randomly distributed
outlying eigenvalues outside the support of the eigenvalue density of $A$, as
previously observed, arises in regions of the complex plane $\Omega$ where
there are nonzero singular values of $L^{-1} (z\mathbf{1} - M) R^{-1}$ (for
$z\in\Omega$) that vanish as $N\rightarrow\infty$. When such singular values do
not exist and $L$ and $R$ are equal to the identity, there is a correspondence
in the normalized Frobenius norm (but not in the operator norm) between the
support of the spectrum of $A$ for $J$ of norm $\sigma$ and the
$\sigma$-pseudospectrum of $M$.
