应用数学学报
應用數學學報
응용수학학보
ACTA MATHEMATICAE APPLICATAE SINICA
2009年
5期
810-818
,共9页
双反对称矩阵%反对称矩阵%Frobenius范数%最佳逼近
雙反對稱矩陣%反對稱矩陣%Frobenius範數%最佳逼近
쌍반대칭구진%반대칭구진%Frobenius범수%최가핍근
anti-bisymmetric matrix%anti-symmetric matrix%frobenius norm%optimal approximation
本文主要讨论下而两个问题并得到相关结果:问题Ⅰ:给定A ∈ R~(k×n),B ∈ R~(k×n),求X ∈ BASR~(n×n),使得AX=B.问题Ⅱ:给定X* ∈R~(n×n),求X使得‖X-X~*‖=minX∈S_E‖X-X~*‖,其中S_E是问题Ⅰ的解集合,‖·‖是Frobenius范数.通过对上述问题的讨论给出了问题Ⅰ解存在的充分必要条件和其解的一般表达式同时给出了问题Ⅱ的解,算法,和数值例子.
本文主要討論下而兩箇問題併得到相關結果:問題Ⅰ:給定A ∈ R~(k×n),B ∈ R~(k×n),求X ∈ BASR~(n×n),使得AX=B.問題Ⅱ:給定X* ∈R~(n×n),求X使得‖X-X~*‖=minX∈S_E‖X-X~*‖,其中S_E是問題Ⅰ的解集閤,‖·‖是Frobenius範數.通過對上述問題的討論給齣瞭問題Ⅰ解存在的充分必要條件和其解的一般錶達式同時給齣瞭問題Ⅱ的解,算法,和數值例子.
본문주요토론하이량개문제병득도상관결과:문제Ⅰ:급정A ∈ R~(k×n),B ∈ R~(k×n),구X ∈ BASR~(n×n),사득AX=B.문제Ⅱ:급정X* ∈R~(n×n),구X사득‖X-X~*‖=minX∈S_E‖X-X~*‖,기중S_E시문제Ⅰ적해집합,‖·‖시Frobenius범수.통과대상술문제적토론급출료문제Ⅰ해존재적충분필요조건화기해적일반표체식동시급출료문제Ⅱ적해,산법,화수치례자.
This paper is mainly concerned with solving the following two problems,Problem Ⅰ: Given k × n real matrices A and B, find X ∈ BASR~(n×n) such that AX = B.Problem Ⅱ: Given an ~(n×n) real matrix X~*, find an n×n matrix X such that ‖X-X~*‖=X ∈S_E‖X-X~*‖,wher‖·‖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 gen-eral solutions of Problem Ⅰ are given. The explicit solution, a numerical algorithm and a numerical example to Problem Ⅱ are provided.
