A note on the complex semi-definite matrix Procrustes problem

This note outlines an algorithm for solving the complex “matrix Procrustes problem”. This is a least-squares approximation over the cone of positive semi-definite Hermitian matrices, which has a number of applications in the areas of Optimization, Signal Processing and Control. The work generalises the method of [J.C. Allwright, “Positive semidefinite matrices: Characterization via conical hulls and least squares solution of a matrix equation”, SIAM J Control and Optimization, 26(3):537-556, 1988], who obtained a numerical solution to the real-valued version of the problem. It is shown that, subject to an appropriate rank assumption, the complex problem can be formulated in a real setting using a matrix dilation technique, for which the method of Allwright is applicable. However, this transformation results in an over-parametrisation of the problem and, therefore, convergence to the optimal solution is slow. Here an alternative algorithm is developed for solving the complex problem, which exploits fully the special structure of the dilated matrix. The advantages of the modified algorithm are demonstrated via a numerical example.