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A note on the best approximate solution of the

equation AXB-C等于0

Song Chuanning, Xu Qingxiang

(Mathematics and Science College,Shanghai Normal University,Shanghai 200234,China)

Abstract:

A counterexample is constructed which shows that with respect to the 2-norm,there exist matrices A,B and C such that A CB fails to be the best approximate solution of the equation AXB-C等于0,where A and B are the Moore-Penrose inverses of A and B,respectively.

Key words:

Moore-Penrose inverse; Frobenius norm; 2-norm

CLC number: O 151.21Document code: AArticle ID: 1000-5137(2018)01-0022-02

摘 要:

舉反例说明:对于矩阵的2-范数,存在矩阵A,B和C,使得ACB不是矩阵方程AXB-C等于0的最佳逼近解,其中A和B分别是A和B的Moore-Penrose逆.

关键词:

Moore-Penrose逆; Frobenius 范数; 2-范数

Received date: 2017-02-25

Foundation item: The National Natural Science Foundation of China (11671261)

Biography: Song Chuanning(1962-),female,associate professor,research area:Matrix and operator theory.E-mail:songning@shnu.edu.cn

引用格式: 宋传宁,许庆祥.关于矩阵方程AXB-C等于0的最佳逼近解的一个注记 [J].上海师范大学学报(自然科学版),2018,47(1):22-23.

Citation format: Song C N,Xu Q X.A note on the best approximate solution of the equation AXB-C等于0 [J].Journal of Shanghai Normal University(Natural Sciences),2018,47(1):22-23.

Throughout this note,Cn×n is the set of n×n complex matrices.For any A∈ Cn×n,let A be its Moore-Penrose inverse[1-2].Let ·F and ·2 be the Frobenius norm and 2-norm (spectral norm) on Cn×n,respectively.

Definition

[2]Let · be any norm on Cn×n,and A,B,C be given in Cn×n.A matrix X0∈Cn×n is said to be the best approximate solution (with respect to the norm ·) of the equation F(X)等于defAXB-C等于0,if for any X∈Cn×n,either F(X)>F(X0),or F(X)等于F(X0) and X≥X0.

It is known from [2,Corollary 1] that with respect to the Frobenius norm ·F,the matrix X0等于ACB is the unique best approximate solution of the equation F(X)等于0.A similar conclusion was asserted in item (b) of [1,Theorem 2.1] for the 2-norm.In this note,we will show that item (b) of [1,Theorem 2.1] is actually incorrect.Our counterexample is as follows:

Example

Let A等于B等于10

00,C等于-1-1

-1-1 and X等于-2x12

x21x22 for any x12,x21,x22∈C.Then

AACBB-C2等于01

112等于1+52,

AXB-C2-11

112等于2,

so AACBB-C2>AXB-C2.

Remark

In view of the counterexample above,the 2-norm A(1,3)Y,KL2 appeared in [1,Lemma 2.3] should be replaced by the Frobenius norm A(1,3)Y,KLF.

References:

[1]Damm T,Stahl D.Linear least squares problems with additional constraints and an application to scattered data approximation [J].Linear Algebra and its Applications,2013,439:933-943.

[2]Penrose R.On best approximate solutions of linear matrix equations [J].Mathematical Proceedings of the Cambridge Philosophical Society,1956,52:17-19.

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