一类矩阵方程的广义双（反）对称解及其最.doc

精品论文一类矩阵方程的广义双（反）对称解及其最小二乘问题王卿文1, 于娟1,21 上海大学数学系，上海 2004442 中国石油大学理学院，青岛 266580摘要：本文给出了方程 ax = b 有广义双对称和广义双反对称解的充要条件及解的表达式. 且 由解的表达式给出了解的极秩. 若方程有解的条件不满足, 给出了其最小二乘广义双对称解和 最小二乘广义双反对称解及极小范数最小二乘广义双对称解和极小范数最小二乘广义双反对称 解. 最后还给出了方程有最小二乘广义双对称解和最小二乘广义双反对称解的算法和算例. 关键词： 广义双（反）对称解，最小二乘解，秩，矩阵方程。中图分类号： o151.1, o151.2.on the generalized bi(skew-)symmetric solutions of a linear matrix equation and its procrust problemswang qing-wen1, yu juan1,21 department of mathematics, university of shanghai, shanghai 2004442 department of basic mathematics, university of china petroleum, qingdao 266580abstract: in this paper, the solvability conditions and the explicit expressions of the generalized bisymmetric and bi-skew-symmetric solutions of the matrix equation ax = b are respectively established by applying two methods. then the maximal and minimal ranks of the solutions are derived. if the solvability conditions are not satised, the generalized bisymmetric and bi-skew-symmetric least squares solutions of the matrix equation are considered, and the generalized bisymmetric and bi-skew-symmetric least squares solutions with the minimum norm are also obtained. in addition, two algorithms are provided to compute the generalized bi(skew-)symmetric least squares solution and an example is given to illustrate that the algorithms are feasible.key words: generalized bi(skew-)symmetric solution, least squares solution, rank, matrixequation.基金项目： this research was supported by the grant from the ph.d. programs foundation of ministry of education ofchina (20093108110001)作者简介： correspondence author：wang qing-wen(1964-), male, full professor，major research direction：linear algebra,matrix theory, operator algebra, linear model. yu juan (1979-), female, lecturer, major research direction：matrix theory.- 28 -0 introductionthe notations and denitions used in this paper are summarized as follows. let cmn , rmn , ornn , srnn and sornn be the sets of all m n complex matrices, all m n real matrices, all n n real orthogonal matrices, all n n real symmetric matrices and all n n real symmetric orthogonal matrices, respectively. the symbols at , a, r(a) and tr(a) respectively stand for the transpose, conjugate transpose, rank and trace of matrix a rmn . in represents the identity matrix of order n. ( a b ) and a b denote a row block matrix and the hadamard product produced by a and b. for two matrices a, b rmn , the inner product is dened by a, b = tr(bt a). obviously, rmn is a complete inner product space. the norm , induced by the inner product, is called the frobenius norm. the moore-penrose generalized inverse a of matrix a rmn , is dened to be the unique solution x rmn satisfying the following four matrix equations(1) ax a = a, (2) x ax = x, (3) (ax ) = ax, (4) (x a) = x a.furthermore, ra and la mean the two orthogonal projectors ra = im aa and la =in aa. it is well known that ra = (ra ) = r2 , la = (la ) = l2 , ra = la andaala = ra .let j = ( en en1 e1 ) with ei the i column of identity matrix in . if j aj = a, then we say that a rnn is centrosymmetric, which has widely practical applications in many areas, such as linear system theory, numerical analysis, information theory and linearestimate theory (see, 1, 2, 3). if a = at = j aj , then a is called the bisymmetric ma- trix. the bisymmetric solutions of the matrix equation(s) has been frequently investigated by many authors; see, e.g. 4, 5, 6. as the extension of bisymmetric matrix, the following two conceptions are dened.denition 0.1. let p sornn , that is, p = p t = p 1 . an n n real matrix a is said to be a generalized bisymmetric matrix with respect to p if a = at = p ap .denition 0.2. let p sornn , that is, p = p t = p 1 . an n n real matrix a is said to be a generalized bi-skew-symmetric matrix with respect to p if a = at = p ap .the set of all n n generalized bisymmetric matrices and the set of all n n generalized bi-skew-symmetric matrices are respectively denoted by gbsrnn and gbssrnn . the gen- eralized bi(skew-)symmetric matrices are very useful in engineering problems, and have been deeply studied by several authors (see, 7, 8).the linear matrix equations have attracted much attention, among which, the well-knownlinear matrix equationax = b (1.1)and its inverse problem have been widely and extensively studied by many authors ( 9, 10, 11,12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22). for instance, gro in 9, lei and cain in 12, and lei in 13 respectively obtained its the explicit solutions, the re-nonnegative denite solutions, and re-positive denite solutions. peng and hu in 15 gave its the reexive and anti-reexive solutions. zhao et al. in 22 got its the bisymmetric least squares solutions under a central principal submatrix constraints.to our knowledge, there is little information on the generalized bisymmetric and bi-skew- symmetric solutions of the equation (1.1) with unknown x . so, in this paper, we rst in section2 establish the necessary and sucient conditions and the explicit expressions of the generalized bisymmetric and bi-skew-symmetric solutions of equation (1.1) by applying two methods. since the extremal ranks, i.e., the maximal and minimal ranks, of some matrix expressions have many applications in statistics, control theory and economics (see, 23, 24, 25, 26, 27), then in section3, we study the maximal and minimal ranks of the generalized bi(skew-)symmetric solutions of equation (1.1). in section 4, we investigate the generalized bi(skew-)symmetric least squares solutions of equation (1.1) by virtue of the singular value decomposition (svd) and denitions1.1 and 1.2. the generalized bisymmetric and bi-skew-symmetric least squares solutions with the minimum norm are corresponding provided, and meanwhile two algorithms and an example are given. finally, in section 5, some conclusions are made.1 the generalized bisymmetric and bi-skew-symmetric solutions of equation (1.1)in order to derive the solvability conditions and the explicit expressions of the bisymmetric and bi-skew-symmetric solutions of equation (1.1), the following lemmas will be required.lemma 1. (28) let a1 rmn , a3 rkn , b2 rrs , b4 rrl , c1 rmr , c2 rns , c3 rkr and c4 rnl be known, x rnr unknown, and k = a3 la , n = rb b4 ,12q = c4 a c1 b4 la c2 bb4 la k q1 n, q1 = c3 a3 a c1 k c2 b. then the system1 12 11 2of matrix equationsa1 x = c1 , x b2 = c2 , a3 x = c3 , x b4 = c4is consistent if and only ifk k q1 rb2 = q1 , qln = 0, rlak q = 0, a c= c b ,1 l 1 2 1 2ai aci = ci , cj bbj = cj , i = 1, 3; j = 2, 4.ijin which case, the general solution of the system can be expressed asx = a c1 + la c2 b + la k q1 rb+ qn rb+ la lk z rn rb ,1 12 12212where z is an arbitrary real matrix with appropriate size.lemma 2. (29) given a cmn , b cpl , c cmp and d cnl . then the system of matrix equationsax = c, x b = dhas a solution x cnp if and only ifaac = c, dbb = d, ad = c b,in which case, the general solutions can be expressed asx = ac + db aadb + (i aa)v (i bb),where v cnp is arbitrary.by denitions 1.1 and 1.2, and the eigenvalue decomposition of the symmetric orthogonal matrix p, it is easily proved that the following two lemmas hold.lemma 3. let the eigenvalue decomposition of p sornn bep = u( ir 00inr)u t ,where u ornn . then x gbsrnn if and only if x has the formx = u( x11 00x22)u t , (2.1)where x11 srrr , x22 sr(nr)(nr) .lemma 4. let the eigenvalue decomposition of p sornn bep = u( ir 00inr)u t ,where u ornn . then x gbssrnn if and only if x has the formx = u(0x12)u t , (2.2)0tx12where x12 rr(nr) .now, we give the general expression of the generalized bi(skew-)symmetric solutions of equation (1.1).theorem 1. given a, b rmn and p sornn . letk = ap la , n = rat p at = la p at , q1 = bp ap ab k bt (at ), q = p bt abp at la bt (at )p at la k q1 n.then the fol lowing statements are equivalent.(1) the equation (1.1) has a solution x gbsrnn .(2) the system of matrix equationsay = b, y at = bt , ap y = bp, y p at = p bt (2.3)has a solution y rnn , in this case, the generalized bisymmetric solution of equation (1.1) isy + y t + p y p + (p y p )tx = . (2.4)4(3) bt (at )at = bt , p bt (p at )(p at ) = p bt , abt = bat ,k k q1 rat = q1 , qln = 0, rla lk q = 0, aab = b, (ap )(ap )bp = bp.in which case, the generalized bisymmetric solutions of equation (1.1) can be expressed asx = 1 ab + (ab)t + p (ab)p + p (ab)t p +4111la bt (at ) + abla4+ p la bt (at )p + p abla p +1la k q1 rat + la qt (k )t la + p la k q1 rat p41+ p la qt (k )t la p +qn rat + la (n )t qt + p qn rat p + p la (n )t qt p 41+ la lk w rn rat + la rn w t lk la + p la lk w rn rat p + p la rn w t lk la p ,4(2.6)where w is an arbitrary real matrix with appropriate size.proof. (1)(2). by denition 1.1, it is not dicult to get that (1) is equivalent to (2). further, in view of (2.4), it can be easily veried that x = x t = p x p . moreover, since y + y t + p y p + (p y p )t a =4=1 (ay + ay t + ap y p + ap y t p )41(b + b + b + b)4= b,then the expression in (2.4) is the generalized bisymmetric solution of equation (1.1). (2)(3). from lemma 2.1, it can be proved that (2) is equivalent to (3). under theconditions of (2.5), the solutions of equation (2.3) can be written asy = ab + la bt (at ) + la k q1 rat + qn rat + la lk w rn rat . (2.7) substituting (2.7) into (2.4) yields (2.6). the proof is completed.similar to the proof of theorem 2.5, the following result is also obtained.theorem 2. given a, b rmn and p sornn . letk = ap la , n = rat p at = la p at , q1 = bp ap ab + k bt (at ), q = p bt abp at + la bt (at )p at la k q1 n.then the fol lowing statements are equivalent.(1) the equation (1.1) has a solution x gbssrnn .(2) the system of matrix equationsay = b, y at = bt , ap y = bp, y p at = p bthas a solution y rnn , in this case, the generalized bi-skew-symmetric solution of equation(1.1) is y y t + p y p (p y p )tx = .4(3) bt (at )at = bt , p bt (p at )(p at ) = p bt , abt = bat ,k k q1 rat = q1 , qln = 0, rla lk q = 0, aab = b, (ap )(ap )bp = bp.in which case, the generalized bi-skew-symmetric solutions of equation (1.1) can be expressed as14x = ab (ab)t + p (ab)p p (ab)t p +41la bt (at ) p la bt (at )p1t t + abla + p abla p + 4 la k q1 rat la q1 (k )1la + p la k q1 rat p 14p la qt (k )t la p +qn rat la (n )t qt + p qn rat p p la (n )t qt p 14+ la lk z rn rat la rn z t lk la + p la lk z rn rat p p la rn z t lk la p ,where z is an arbitrary real matrix with appropriate size.to present the general expressions of the generalized bi(skew-)symmetric solutions of the equation (1.1) by lemmas 2.3 and 2.4, the following partitions are needed. according to the eigenvalue decomposition of the symmetric orthogonal matrixpartitionp = u( ir 00inr)u t ,12au = ( aa ) , a1 rmr , a2 rm(nr) , (2.8)12bu = ( bb ) , b1 rmr , b2 rm(nr) . (2.9)let the svds of a1 and a2 be respectively1a1 = u1v t , (2.10)( 10 )00( 20 )002a2 = u2v t , (2.11)where1 = diag(1, , r1 ) 0, r1 = r(a1 ), 2 = diag(1, , r2 ) 0, r2 = r(a2 ).putv tv t1 x11 v1 =12( x11t) t, x11 = x11t, x22 = x2211(1, u t b1 v1 =)12 , (2.12)x12x22b21 b22g( xg11x12 ), xt= x, xt= x, u t b v( bg11bg12), (2.13)2 x22 v2 =txg12xg22g11g11g22g222 2 2bg21bg22compatible with (2.10) and (2.11), respectively. combining (2.1) and the equation (1.1) yields that equation (1.1) has a solution x gbsrnn if and only if there exist x11 srrr and x22 sr(nr)(nr) such thata1 x11 = b1 , a2 x22 = b2 .further, there exists x11 srrr such thata1 x11 = b1if and only ifa1 bt = b1 at(2.14)1 1andminx11 srrrr(a1 x11 b1 ) = 0, (2.15)and there exists x22 sr(nr)(nr) such thata2 x22 = b2if and only ifa2 bt = b2 at(2.16)andmin2 2r(a2 x22 b2 ) = 0. (2.17)x22 sr(nr)(nr)it follows from (2.10), (2.12) and (2.14) thatt1 b11= b11 1 ,that is,1t1 b11 = b1111 .combining (2.10), (2.12) and (2.15) yields that( 1 0 ) ( x11 12 )()1112r(a1 x11 b1 ) = r( 00x12t x22b21 )b22= r( 1 x11 b11112 )12 )= 0.so, there exists x11 srrr such thatb21b22a1 x11 = b1if and only ifand1t1 b11 = b1111 , b21 = 0, b22 = 0, (2.18)x11 = 1 b11 , x12 = 1 b12 .similarly,and1t21 bg11 = bg112 ,11bg21 = 0,bg22 = 0, (2.19)xg11 = 1 b11 ,x12 = 1 b122 g g2 gcan be obtained from (2.11), (2.13), (2.16) and (2.17). hence the general expression of the generalized bisymmetric solutions of equation (1.1) can be expressed as follows.theorem 3. given a, b rmn and p sornn . let the partitions of au and bu , thesvds of a1 and a2 be (2.8)-(2.11). v t x11 v1 , u t b1 v1 , v t x22 v2 and u t b2 v2 have the forms1 1 2 2as in (2.12) and (2.13). then equation (1.1) has a solution x gbsrnn if and only if(2.18) and (2.19) hold. when these conditions are satised, the solution x can be written aswherex = u( x11 00x22)u t ,( 1 b11 1 12 )(122bg111bg12 )x11 = v11t1b12111x22v t , x22 = v2t22bg121xg22v t ,and x22 sr(rr1 )(rr1 ) , xg22 sr(nrr2 )(nrr2 ) are arbitrary.combining (2.2), (2.8) and (2.9) yields that equation (1.1) has a solution x gbssrnnif and only if the system of matrix equationsa1 x12 = b2 , x12 at = bt(2.20)2 1is consistent. then by lemma 2.2, there exists x12 rr(nr) such that (2.20) holds if andonly ifa1 a b2 = b2 , a2 a b1 = b1 , a1 bt = b2 at . (2.21)1 2 1 2when all equalities in (2.21) hold, the solution x12 can be expressed asx12 = a b2 bt (a )t + a a1 bt (a )t + la w la , (2.22)1121 1 2 12where w is an arbitrary real matrix with appropriate size. thus we have the following theorem.theorem 4. given a, b rmn and p sornn . let the partitions of au and bu be respectively as in (2.8) and (2.9). then equation (1.1) has a solution x gbssrnn if and only if al l equalities in (2.21) hold, in this case, the solution x can be described asx = u(0x12)u t ,x0t12where x12 has the form as in (2.22).2 the extremal ranks of the generalizedbi(skew-)symmetric solutions of equation (1.1)in this section, the following lemmas will be needed to investigate the maximal and minimal ranks of the solution x gbsrnn (x gbssrnn ) of the equation (1.1).lemma 5. (30) letm (x12 ) =( a11 x12a21 a22), a11 cs1 t1 , a21 cs2 t1 , a22 cs2 t2 .then(1) the maximal rank of the matrix m (x12 ) ismaxr(m (x12 ) = minx12 ( a11r(a21)21) + t2 , r( aa22) + s1with21x12 = xd12 + a11 a a22 + a11 la21 v + w ra21a22 .(2) the minimal rank of the matrix m (x12 ) ismin(m (x12 ) = r(x12( a11a21)21) + r( aa22) r(a21 )withx12 = a11 a a22 + a11 laa2121 21 v + w ra22 .where v, w are arbitrary matrices with appropriate size, and xd12 satisesr(rg xd12 lh ) =