Abstract and Applied Analysis

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 ${\sum }_{k=1}^{p}{\parallel {A}_{k}X-{C}_{k}\parallel }^{2} + {\sum }_{l=1}^{q}{\parallel X{B}_{l}-{D}_{l}\parallel }^{2}$ 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.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 423605, 9 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393447686

Digital Object Identifier
doi:10.1155/2013/423605

Mathematical Reviews number (MathSciNet)
MR3108633

Zentralblatt MATH identifier
1291.15035

Citation

Chang, Hai-Xia; Duan, Xue-Feng; Wang, Qing-Wen. The Hermitian $R$ -Conjugate Generalized Procrustes Problem. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 423605, 9 pages. doi:10.1155/2013/423605. https://projecteuclid.org/euclid.aaa/1393447686