On the non-symmetric semidefinite Procrustes problem

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