Vo136(2016) No3 数学杂志 THE EXTREMAL RANK SoLUT10NS oF THE MATRIX EQUATIONS AX=B， =D XIAO Qingfeng (Department ol Basic，Dongguan Polytechnic，Dongguan 523808，China) Abstract：In this paperwe considered the rank range of the solutions of a class of matrix equationsBy applying the singular value decomposition of matrix and the properties of Frobenius matrix norm we obtained the extremal rank and the solution expression of under rank constrained Some special cases of theses problems are considered，and some results are obtained Keywords： optimal control；extremal rank；SVD decomposition；Probenius matrix norm 2010 MR Subject Classification： 65F05；65F18 Document code： A Article ID：02557797(2016103045807 1 Introduction 、ve first introduce some notations to be usedLet C denote the set of all nm complex matrices；R denote the set of all礼m real matrices；OR be the sets of all n x n orthogonal matricesThe symbols A ，A+，A一，R(A)，N(A)and r(A)stand for the transpose，MoorePenrose generalized inverse，any generalized inverse，range(column space)，null space and rank of AR ，respectivelyThe symbols EA and FA stand for the two projectors EA=I Aand =I一 一A induced by AThe matrices I and 0 respectively,denote the identity and zero matrices of sizes implied by context use ( ，B)=trace(B A)to define the inner product of matrices A and B in R Then R is a Hilbert inner product spaceThe norm of a matrix generated by the inner product is the Frobenius norm l11l，that is IIAIl=、( ，A)=(trace(AT ) Researches on extreme ranks of solutions to linear matrix equations was actively ongoing for more than 30 yearsFor instance，Mitra1considered solutions with fixed ranks for the matrix equations Ax=B and AxB=c；Mitra2gave common solutions of minimal rank of the pair of matrix equations Ax=C， B=D；Uhlig【3gave the maximal and minimal ranks of solutions of the equation AX：B；Mitra I41 examined common solutions of minimal rank of the pair of matrix equations A1X1Bl=Ci and A2X2B2=C2By applying the matrix rank method，recently，Tian51 obtained the minimal rank of solutions to the matrix equation A=JE +yCIn 2003，Tian in6，7investigated the extremal ranks solutions to 恤e complex matrix equation B=C and gave some applicationsIn 2006，Lin and Wang Received date：20131008 Foundation item：Supported by Biography：Xiao Qingfeng(1977一 in matrix theory and applications Accepted date：2013-12-10 Scientific Research Fund of Dongguan Polytechnic(2012a03) )j male，born at Loudi，Hunan，associate professor，doctor，major No3 The extremal rank s0lutions of the matrix equations AX=B，XC=D 459 in Isstudied the extreme ranks of solutions to the system of matrix equations A1 X=C1， XB2： ，A3XB3：C3 over an arbitrary division ring，which was investigated in9and 10Recently，Xiao et a1considered the extremal ranks，iemaximal and minimal ranks to the equation AX=B(see，eg1115】) In this paper，we consider the extremal rank solutions of the matrix equations AX=BXC=D where ARpm，BRp ，CR”q，D are given matrices The paper is organized as followsAt first，we will introduce several lemmas which wm be used in the latter sectionsIn Section 3，applying the matrix rank method，we will discuss the rank of the general solution to the matrix equations =B，XC：D，where ARpm，BRp ，CR q，DR are given matrices- 2 Some Lemmas LemlTla 21 see6)Let A，B，C，and D be m ，mk，l他，l matrices， respectivelyThen r =rcA + ccc，一AA ， A暑= +r B 一r c ，+r cEa(D-CA-B) (22) where G=CFA and日=EAB LemlTla 22 see16) Given ARp ，BRpXn CRnx口，DR 。Let the singular value decompositions of A be， = E： T= ， (23) where U：(U1， 1ORpp，U1Rp ， =(， )OR ， R ，k=r( ) ：diag( 1， 2， )， 1 0Let the singular value decomp OSitions of B be, ooQv=PlrQ (24) where P：( ，尸2)ORnn，P1R ，Q=(Q1，Q2)OR。 ，Q1R。 ，t=r( )， r：diag( 1， ， )，71 0Then the matrix equations(11)have a solution in R if and only if BC=ADAA+B=B，DC+C=C Moreover，its general solution can be expressed as X=DC+A+BA+ADC+(，一A+A)Z(ICC+)，VzR (25) (26) where vT x P：X I f27) 妻 V 。： ：一 B =啦Pl 2TDQ rl m i 三三 鲫 ， is arbitrary t。P眠ro。omfittheBy(2州6),Z is arbitrary)we from(2叭2-4jand(27)that rary 3 The EXtremal Rank s。Iuti0ns t0(11) can bAeswrsumj e thaesmatrix equations(11)has a s。lu ，and the generaIs0lu where 乏 X l 二 BP1= Pl，五。：一 JE l= 3 P-1VfDQ ：哆 。 Let G1： l H be respectivev v,Txp2 11 2，AsSume the 8jngular va1ue dec0mposition0fGl and G ( )曙 皤， where Uc谍 ，J-l Eiag rn- k)k VG1 l ， ： (G )， ：d ( ， ，、 ( ， 。) 研 (， o)Q 1+=PllFIQ五 (32) OR七 (33) Qu#=(Q1l，Q12)ORk 。 l0 旧 呲 里 n 眦 嘁 聃砌三j、 m ， 咖莹眦 L 嘣 、 2 2 ) 1、J I 怕 n ，n 、三 ， 功 ” e 一 ： 2+ H 研 ， 七 No3 The extremal rank solutions of the matrix equations AX=BXC=D 461 Theorem 31 Given ARp ，BRp”，CR”q，DR qthe singular value decompositions of the matrices A，C and GI，H are given by(23)，(24)and(32)，(33)， respectivelyThen equations(11)has a solution if