The Hermitian R -Conjugate Generalized Procrustes Problem

Abstract

We consider the Hermitian $R$-conjugate generalized Procrustes problem to find Hermitian $R$-conjugate matrix $X$ such that ${\sum }_{k=1}^p{\parallel A_{k}X-C_k\parallel }^2\hspace{0.17em}+\hspace{0.17em}{\sum }_{l=1}^q{\parallel X{B}_l-D_l\parallel }^2$ is minimum, where $A_k$, $C_k$, $B_l$, and $D_l$ ($k=\mathrm{1,2},\dots ,p$, $l=\mathrm{1},\dots ,q$) 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 $A_{\mathrm{1}}X=C_{\mathrm{1}}$, $A_{\mathrm{2}}X=C_{\mathrm{2}},\dots ,A_{p}X=C_p$, $X{B}_{\mathrm{1}}=D_{\mathrm{1}},\dots ,X{B}_q=D_q$ ($p$ 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.