The Hermitian R-Conjugate Generalized Procrustes Problem

الملخص EN

We consider the Hermitian R-conjugate generalized Procrustes problem to find Hermitian R-conjugate matrix X such that ∑k=1p∥AkX-Ck∥2 + ∑l=1q∥XBl-Dl∥2 is minimum, where Ak, Ck, Bl, and Dl (k=1,2,…,p, l=1,…,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 A1X=C1, A2X=C2,…,ApX=Cp, XB1=D1,…,XBq=Dq (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.