## The Hermitian $$R$$-conjugate generalized Procrustes problem.(English)Zbl 1291.15035

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.

### MSC:

 15A24 Matrix equations and identities 65F30 Other matrix algorithms (MSC2010) 65K10 Numerical optimization and variational techniques 90C20 Quadratic programming
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