An N×N iterative photonic processor is proposed for the first time, we believe, for fast computation of complex-valued matrix inversion, a fundamental but computationally expensive linear algebra operation. Compared to traditional digital electronic processing, optical signal processing has a few unparalleled features that could enable higher representational efficiency and faster computing speed. The proposed processor is based on photonic integration platforms–the inclusion of III-V gain blocks offers net neutral loss in the phase-sensitive loops. This is essential for the Richardson iteration method that is adopted in this paper for complex linear systems. Wavelength multiplexing can be used to significantly improve the processing efficiency, allowing the computation of multiple columns of the inverse matrix using a single processor core. Performances of the key building blocks are modeled and simulated, followed by a system-level analysis, which serves as a guideline for designing an N×N Richardson iteration processor. An inversion accuracy of >98% can be predicted for a 64×64 photonic processor with a >80 times faster inversion rate than electronic processors. Including the power consumed by both active components and electronic circuits, the power efficiency of the proposed processor is estimated to be over an order of magnitude more energy-efficient than electronic processors. The proposed iterative photonic integrated processor provides a promising solution for future optical signal processing systems.