Matrix factorizations are important in many real-time signal processing applications. In order to improve the performance of these algorithms, special purpose VLSI processor arrays are being developed. Recently, the Coordinate Rotation Digital Computer (CORDIC) algorithms have been applied to the QR Decomposition (QRD) and the Singular Value Decomposition (SVD). In this paper, the CORDIC arithmetic algorithms are extended to deal with complex data. Novel CORDIC VLSI architectures for the QRD of a complex matrix, the Eigenvalue Decomposition of a Hermitian matrix, and the SVD of a complex matrix are presented. These architectures are suitable for VLSI implementation.