The Hermitian \(R\)-conjugate generalized Procrustes problem (Q2015531)

Summary: We consider the Hermitian \(R\)-conjugate generalized Procrustes problem to find Hermitian \(R\)-conjugate matrix \(X\) such that \(\sum^p_{k=1}||A_kX-C_k||^2+\sum^q_{l=1}||XB_l-D_l||^2\) is minimum, where \(A_k\), \(C_k\), \(B_l\), and \(D_l(k=1,2,\dots,p,l=1,\dots,q)\) are given complex matrices, and \(p\) and \(q\) are positive integers. The expression of the solution to Hermitian \(R\)-conjugate generalized Procrustes problem is derived. And the optimal approximation solution in the solution set for Hermitian \(R\)-conjugate generalized Procrustes problem to a given matrix is also obtained. Furthermore, we establish necessary and sufficient conditions for the existence and the formula for Hermitian \(R\)-conjugate solution to the linear system of complex matrix equations \(A_1X=C_1\), \(A_2X=C_2,\dots,A_pX=C_p\), \(XB_1=D_1,\dots,XB_q=D_q\) (\(p\) and \(q\) are positive integers). The representation of the corresponding optimal approximation problem is presented. Finally, an algorithm for solving two problems above is proposed, and the numerical examples show its feasibility.