On the non-symmetric semidefinite Procrustes problem

Abstract

In this paper, we consider the non-symmetric positive semidefinite Procrustes (NSPSDP) problem: Given two matrices X,B∈Rn,m, find the matrix A∈Rn,n that minimizes the Frobenius norm of AX−B and which is such that A+AT is positive semidefinite. We generalize the semi-analytical approach for the symmetric positive semidefinite Procrustes problem, where A is required to be positive semidefinite, that was proposed by Gillis and Sharma (A semi-analytical approach for the positive semidefinite Procrustes problem, Linear Algebra Appl. 540, 112-137, 2018). As for the symmetric case, we first show that the NSPSDP problem can be reduced to a smaller NSPSDP problem that always has a unique solution and where the matrix X is diagonal and has full rank. Then, an efficient semi-analytical algorithm to solve the NSPSDP problem is proposed, solving the smaller and well-posed problem with a fast gradient method which guarantees a linear rate of convergence. This algorithm is also applicable to solve the complex NSPSDP problem, where X,B∈Cn,m, as we show that the complex NSPSDP problem can be written as an overparametrized real NSPSDP problem. The efficiency of the proposed algorithm is illustrated on several numerical examples.