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Matrix-variate optimization plays a central role in advanced wireless systemdesigns. In this paper, we aim to explore optimal solutions of matrix variablesunder two special structure constraints using complex matrix derivatives,including diagonal structure constraints and constant modulus constraints, bothof which are closely related to the state-of-the-art wireless applications.Specifically, for diagonal structure constraints mostly considered in theuplink multi-user single-input multiple-output (MU-SIMO) system and theamplitude-adjustable intelligent reflecting surface (IRS)-aided multiple-inputmultiple-output (MIMO) system, the capacity maximization problem, themean-squared error (MSE) minimization problem and their variants are rigorouslyinvestigated. By leveraging complex matrix derivatives, the optimal solutionsof these problems are directly obtained in closed forms. Nevertheless, forconstant modulus constraints with the intrinsic nature of element-wisedecomposability, which are often seen in the hybrid analog-digital MIMO systemand the fully-passive IRS-aided MIMO system, we firstly explore inherentstructures of the element-wise phase derivatives associated with differentoptimization problems. Then, we propose a novel alternating optimization (AO)algorithm with the aid of several arbitrary feasible solutions, which avoidsthe complicated matrix inversion and matrix factorization involved inconventional element-wise iterative algorithms. Numerical simulations revealthat the proposed algorithm can dramatically reduce the computationalcomplexity without loss of system performance.

A Framework on Complex Matrix Derivatives with Special Structure Constraints for Wireless Systems