IJPAM: Volume 52, No. 5 (2009)

THE ANTI-BISYMMETRIC MATRICES OPTIMAL
APPROXIMATION SOLUTION OF
MATRIX EQUATION $AX=B$

Zhang Xindong$^1$, Zheng Chengmin$^2$, Zhang Zhinan$^3$
$^{1,2}$College of Mathematical Physics and Information Sciences
Xinjiang Normal University
Urumqi, Xinjiang, 830054, P.R. CHINA
$^1$e-mail: liaoyuan1126@163.com
$^2$e-mail: zcm19940809@sina.com
$^3$College of Mathematics and System Sciences
Xinjiang University
Urumqi, Xinjiang, 830046, P.R. CHINA
e-mail: znz01@xju.edu.cn


Abstract.This paper is mainly concerned with solving the following two problems:

Problem I. Given $k\times n$ real matrices $A$ and $B$, find $X\in BASR^{n\times n}$ such that $AX=B.$

Problem II. Given an $n\times n$ real matrix $X^{\ast}$, find an $n\times n$ matrix $\widehat{X}$ such that $\Vert\widehat{X}-X^{\ast}\Vert={\min\limits_{X\in S_{E}}}\Vert X-X^{\ast}\Vert$, where $\Vert\cdot\Vert$ is a Frobenius norm, and $S_{E}$ is the solution set of Problem I.

The necessary and sufficient conditions for the existence and expressions of the general solutions of Problem I are given. The explicit solution, a numerical algorithm and a numerical example to Problem II are provided.

Received: May 15, 2008

AMS Subject Classification: 15A52

Key Words and Phrases: anti-bisymmetric matrices, anti-symmetric matrix, Frobenius norm, optimal approximation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 52
Issue: 5