  We investigate the microcanonical version of the complex induced Ginibre
ensemble, by introducing a fixed trace constraint for its second moment. Like
for the canonical Ginibre ensemble, its complex eigenvalues can be interpreted
as a two-dimensional Coulomb gas, which are now subject to a constraint and a
modified, collective confining potential. Despite the lack of determinantal
structure in this fixed trace ensemble, we compute all its density correlation
functions at finite matrix size and compare to a fixed trace ensemble of normal
matrices, representing a different Coulomb gas. Our main tool of investigation
is the Laplace transform, that maps back the fixed trace to the induced Ginibre
ensemble. Products of random matrices have been used to study the Lyapunov and
stability exponents for chaotic dynamical systems, where the latter are based
on the complex eigenvalues of the product matrix. Because little is known about
the universality of the eigenvalue distribution of such product matrices, we
then study the product of $m$ induced Ginibre matrices with a fixed trace
constraint - which are clearly non-Gaussian - and $M-m$ such Ginibre matrices
without constraint. Using an $m$-fold inverse Laplace transform, we obtain a
concise result for the spectral density of such a mixed product matrix at
finite matrix size, for arbitrary fixed $m$ and $M$. Very recently local and
global universality was proven by the authors and their coworker for a more
general, single elliptic fixed trace ensemble in the bulk of the spectrum.
Here, we argue that the spectral density of mixed products is in the same
universality class as the product of $M$ independent induced Ginibre ensembles.
