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Chang, Hai-Xia; Duan, Xue-Feng; Wang, Qing-Wen

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

Abstr. Appl. Anal. 2013, Article ID 423605, 9 p. (2013).
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.

Cited in 1 Document

MSC:

15A24 Matrix equations and identities
65F30 Other matrix algorithms (MSC2010)
65K10 Numerical optimization and variational techniques
90C20 Quadratic programming
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\textit{H.-X. Chang} et al., Abstr. Appl. Anal. 2013, Article ID 423605, 9 p. (2013; Zbl 1291.15035)
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