  Programmable photonic integrated circuits represent an emerging technology
that amalgamates photonics and electronics, paving the way for light-based
information processing at high speeds and low power consumption. Considering
their wide range of applications as one of the most fundamental mathematical
operations there has been a particular interest in programmable photonic
circuits that perform matrix-vector multiplication. In this regard, there has
been great interest in developing novel circuit architectures for performing
matrix operations that are compatible with the existing photonic integrated
circuit technology which can thus be reliably implemented. Recently, it has
been shown that discrete linear unitary operations can be parameterized through
diagonal phase parameters interlaced with a fixed operator that enables
efficient photonic realization of unitary operations by cascading phase shifter
arrays interlaced with a multiport component. Here, we show that such a
decomposition is only a special case of a much broader class of factorizations
that allow for parametrizing arbitrary complex matrices in terms of diagonal
matrices alternating with a fixed unitary matrix. Thus, we introduce a novel
architecture for physically implementing discrete linear operations. The
proposed architecture is built on representing an $N \times N$ matrix operator
in terms of $N+1$ amplitude-and-phase modulation layers interlaced with a fixed
unitary layer that could be implemented via a coupled waveguide array. The
proposed architecture enables the development of novel families of programmable
photonic circuits for on-chip analog information processing.
