  We introduce and study `matricial circular systems' of operators which play
the role of matricial counterparts of circular operators. They describe the
asymptotic joint *-distributions of blocks of independent block-identically
distributed Gaussian random matrices with respect to partial traces. Using
these operators, we introduce `circular free Meixner distributions' as the
non-Hermitian counterparts of free Meixner distributions and construct for them
a random matrix model. Our approach is based on the concept of matricial
freeness applied to operators on Hilbert spaces. It is closely related to
freeness with amalgamation over the algebra A of r x r diagonal matrices
applied to operators on Hilbert A-bimodules.
