## 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

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