(R,S)最小化问题.pdfVIP

Linear Algebra and its Applications 389 (2004) 23–31 /locate/laa Minimization problems for (R, S)-symmetric and (R, S)-skew symmetric matrices William F. Trench∗ Trinity University, San Antonio, TX 78212-7200, USA Received 14 October 2003; accepted 24 March 2004 Submitted by V. Mehrmann Abstract Let R ∈ Cm ×m and S ∈ Cn ×n be nontrivial involutions; thus R = R−1 =/ ±I and S = S−1 =/ ±I . We say that A ∈ Cm ×n is (R, S)-symmetric ((R, S)-skew symmetric) if RAS = A (RAS = −A). Let S be the class of m × n (R, S)-symmetric matrices or the class of m × n (R, S)-skew symmetric matrices. Let Z ∈ Cn ×q and W ∈ Cm ×q . We study the following problems: (i) Give necessary and sufficient conditions for the existence of an A ∈ S such that AZ = W , and find all such matrices if the conditions are met. (ii) Find σ (Z, W ) = minA∈S AZ − W and characterize the class S(Z, W ) = {A ∈ S| AZ − W = σ (Z, W )}. (iii) If B ∈ Cm ×n is arbitrary, find σ (Z, W, B) = minA∈S(Z,W) A − B and find A ∈ S(Z, W ) such that A − B = σ (Z, W, B). We obtain explicit formulas for σ (Z, W ), σ (B, Z, W ), and all the matrices in question. © 2004 Elsevier Inc. All rights reserved. AMS classification: 15A24; 15A29 Keywords: Approximation; Frobenius norm; Involution; Moore–Penrose inverse; (R, S)-skew symmet- ric; (R, S)-symmetric 1. Introduction Throughout this paper R ∈ Cm ×m and S ∈ Cn ×n are nontrivial involutions; thus R = R−1 =/ ±I and S = S−1 =/ ±I . We say that A ∈ Cm ×n is (R, S)-symmetric ∗ Mailing address: 95 Pine Lane, Woodland Park, CO 80863, USA. Tel.: +1-719-687-2109. E-mail address: wtrench@ (W.F. Trench). 0024-3795/$ - see fr