  This paper establishes that generic linear quantum stochastic systems have a
pure cascade realization of their transfer function, generalizing an earlier
result established only for the special class of completely passive linear
quantum stochastic systems. In particular, a cascade realization therefore
exists for generic active linear quantum stochastic systems that require an
external source of quanta to operate. The results facilitate a simplified
realization of generic linear quantum stochastic systems for applications such
as coherent feedback control and optical filtering. The key tools that are
developed are algorithms for symplectic QR and Schur decompositions. It is
shown that generic real square matrices of even dimension can be transformed
into a lower $2 \times 2$ block triangular form by a symplectic similarity
transformation. The linear algebraic results herein may be of independent
interest for applications beyond the problem of transfer function realization
for quantum systems. Numerical examples are included to illustrate the main
results. In particular, one example describes an equivalent realization of the
transfer function of a nondegenerate parametric amplifier as the cascade
interconnection of two degenerate parametric amplifiers with an additional
outcoupling mirror.
