一四元数矩阵方程组的 η-厄尔米特解和双.doc

精品论文一四元数矩阵方程组的 -厄尔米特解和双-厄尔米特解 王卿文，何卓衡 上海大学理学院，上海 200444摘要：设 hmn 是实四元数代数 h = a0 + a1 i + a2 j + a3 k i2 = j2 = k2 = ijk = 1, a0 , a1 , a2 , a3 r上的所有 m n 矩阵. 四元数 -厄尔米特矩阵 a 满足 a = a, 双 -厄尔米特矩阵满足a = a = a() , 其中 i, j, k. 本文主要研究如下的实四元数矩阵方程组有 -厄尔米特解的充分必要条件及其通解表达式a1 x = c1 , x b1 = d1 , a2 y = c2 , y b2 = d2 , c3 x c + d3 y d = a3 .3 3作为应用，给出了如下实四元数矩阵方程组存在双 -厄尔米特解的充分必要条件及其通解表达式cabx = cb, x bb = db, ccx c = a3 .相关算法及其数值例子也给出。关键词：矩阵方程组, 四元数矩阵, 广义逆, 双 -厄尔米特矩阵中图分类号： o151.1，o151.2the -hermitian and -bihermitian solutions to some systems of real quaternion matrix equationswang qing-wen, he zhuo-hengdepartment of mathematics, shanghai university, shanghai 200444abstract: let hmn be the set of all m n matrices over the real quaternion algebra h = a0 + a1 i + a2 j + a3 k i2 = j2 = k2 = ijk = 1, a0 , a1 , a2 , a3 r. a = (aij ) hmn,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-），e-mail: ，full professor and super-visor of phd students, major research direction：matrix theory, linear algebra, operator algebra, statistics. he zhuo-heng（1987-），，major research direction：matrix theory.- 29 -a = (aji) hnm, a() = (amj+1,ni+1 ) hnm, where ajiis the conjugate of thequaternion aji. we call that a hnn is -hermitian if a = a, i, j, k; a hnn is -bihermitian if a = a = a() . we in this paper consider a system of linear real quaternion matrix equations involving -hermicity, i.e.a1 x = c1 , x b1 = d1 , a2 y = c2 , y b2 = d2 ,c3 x c + d3 y d = a3 . (1)3 3we present some necessary and sucient conditions for the existence of the -hermitian solution to the system of linear real quaternion matrix equations (1) and give an expression of the -hermitian solution to system (1) when it is solvable. as an application, we consider the necessary and sucient conditions for the systemcabx = cb, x bb = db, ccx c = a3 (2)to have an -bihermitian solution. we establish an expression of the -bihermitian to system(2) when it is solvable. we also obtain a criterion for a quaternion matrix to be-bihermitian. moreover, we provide an algorithm and a numerical example to illustrate the theory developed in this paper.key words: system of quaternion matrix equations; -hermitian solution; -bihermitian solution; quaternion involutions; moore-penrose inverse.0 introductionthroughout this paper, let r, c, and hmn stand, respectively, for the real number eld, the complex number eld, and the set of all m n matrices over the real quaternion algebra h = a0 + a1 i + a2 j + a3 k i2 = j2 = k2 = ijk = 1, a0 , a1 , a2 , a3 r.for a hmn, the symbols a and r(a) denote the conjugate transpose and the rank of a, respectively. the identity matrix with appropriate size is denoted by i . the moore-penrose inverse of a hmn, denoted by a, is dened to be the unique solution x to the following four matrix equations(1) ax a = a, (2) x ax = x,(3) (ax ) = ax,(4) (x a) = x a.furthermore, la and ra stand for the two projectors la = i aa and ra = i aa inducedaaby a, respectively. it is known that la = land ra = r . the quaternion matrix involutionsare dened as a = a, i, j, k. some useful properties of these involutions can be found in 1 and 2(where a = a).we denote the n n permutation matrix whose elements along the southwest-northeastdiagonal are ones and whose remaining elements are zeros by vn. by 1 and 3, we have the following denitions.denition 0.1. (1) a hnn is cal led -hermitian if a = a, i, j, k.denition 0.2. (3) let a = (aij ) hmn, a() = (amj+1,ni+1 ) hnm, a = (ami+1,nj+1 ) hmn, where aji is the conjugate of the quaternion aji. we cal l that a hmn is centrosym- metric if a = a; a hnn is bisymmetric if a = a = a() .denition 0.3. a hnn is cal led -bihermitian if a = a = a() , where a() = a() , i, j, k.clearly, -bihermitian is more general than bisymmetric.the set of all n n -bihermitian matrices is denoted by bn.remark 0.1. (1) for any matrices a hmn and b hnp over h, it is easy to verify thata() = vnavm,a() = vnavm,vna() vm = a,vna() vm = a,(ab)() = b() a() , (ab)() = b() a() ,(ab) = ab,(a) = (a),(a() ) = (a)() , (a() ) = (a)() ,andif a is invertible.(a1 )() = (a() )1 , (a1 )() = (a() )1 , (a1 ) = (a)1(2) clearly, vn = v () = v = v 1 = v = v () .n n n n nquaternions were introduced by irish mathematician sir william rowan hamilton in 1843; they are associative and noncommutative. general properties of matrices over h can be found in 4. quaternions have found a huge amount of practical applications and have been considered in contemporary mathematics such as associative algebras, analysis, topology, etc. nowadays quaternion matrices play an important role in computer science, quantum physics, signal and color image processing, and so on (e.g. 5-10).linear matrix equations have been one of the topics of very active research in matrix theory and applications, and a lot of papers have presented several methods for solving several matrix equations (e.g. 11-31). in mathematics, engineering, and others, many problems can be transformed into some linear matrix equations. for instance, the growth curve model in statistics is consistent if and only if the matrix equationax b + c y d = eis solvable 32. note that the above equation was investigated by many authors, such asbaksalary and kala 33, huang et al. 20, zgler 34, wang et al. 23, and the others.moreover, a recent paper 25 by wang et al. gave the solvability conditions for the existenceof the general solution to the systema1 x = c1 , x b1 = d1 ,a2 y = c2 , y b2 = d2 ,a3 x b3 + a4 y b4 = ccover h, and obtained an expression of the general solution to the above system when the solvability conditions are satised.the hermitian solutions to some matrix equations were investigated in many papers. kha- tri and mitra 21 gave necessary and sucient conditions for the existence of the hermitiansolution to the equationsax = b (3)anda1 x = c1 , x b1 = d1 , (4)respectively, and presented the explicit expressions for the general hermitian solutions to (3) and (4) based on generalized matrix inverses and the matrix rank. for operator equations, phadke and thakare 35 considered the common hermitian solution to (4) for hilbert space operators. daji and koliha in 36 presented the formula for the general hermitian solution to (4) in rings. xu in 37 investigated the solvability conditions for (4) to have a common hermitian solution in the framework of hilbert c -modules, and gave the general hermitian solution to (4) when it is solvable. the hermitian solution to the matrix equationax a + by b = c (5)has been investigated by many authors from dierent approaches. for example, liao and bai 38 arrived at the symmetric positive semidenite solution to (5) by generalized singular value decomposition. xu et al. 39 gave the general form of all least-squares hermitian (skew- hermitian) solutions to (5). deng and hu 40 considered the hermitian and nonnegative solutions to (5) by making use of the quotient singular value decompositions of matrices. liu in 41 obtained the general hermitian solutions to (4). in 2012, farid, moslehian, wang and wu 17 considered some necessary and sucient conditions for the existence of a hermitian solution to the system of equationsa2 y = c2 , y b2 = d2 , a3 z = c3 , z b3 = d3 ,c4 y c + d4 z d = a4 (6)4 4for adjointable operators between hilbert c-modules, and provided an expression for the gen-eral hermitian solution to the system (6). in 2012, dong, wang and zhang 42 gave necessaryand sucient conditions for the existence of the common positive solution to the system (6) for adjointable operators between hilbert c-modules, and presented an expression of the positive solution to the system (6) when the solvability conditions are satised.the -hermitian matrices arise in widely linear modelling, convergence analysis in statis- tical signal processing (e.g. 43-45). recently, horn and zhang 2 provided an analogous special singular value decomposition for -hermitian matrices. yuan and wang 46 derived the expressions of the least squares -hermitian solution of the real quaternion matrix equation ax b + c x d = e. very recently, he and wang 18 considered a real quaternion matrix equation involving -hermicity, i.e.,a1 x + (a1 x ) + b1 y b + c1 z c = d1 ,1 1where y and z are required to be -hermitian. he and wang 18 provided some necessary and sucient conditions for the existence of a solution (x, y, z ) to the equation and presented a general solution when the equation is solvable. they also investigated some properties of the quaternion matrix a in 18.centrosymmetric and bisymmetric matrices have been widely discussed since 1939, which are very useful in engineering problems, information theory, linear system theory, linear esti- mation theory and numerical analysis theory, and others (e.g., 3, 19, 47, 48). wang in 49 derived the bisymmetric solution to system (4).note that the system of matrix equations (6) is a special case of the following system of real quaternion matrix equationsa1 x = c1 , x b1 = d1 , a2 y = c2 , y b2 = d2 ,c3 x c + d3 y d = a3 , (7)3 3to our knowledge, there has been little information on both the -hermitian solution to the system (7) and the -bihermitian solution to systemcabx = cb, x bb = db, ccx c = a3 (8)so for. motivated by the work mentioned above and keeping the interest and wide application of -hermitian matrices, we consider the -hermitian solution to system (7) and use the results to investigate the -bihermitian solution to system (8).the rest of this paper is organized as follows. we in section 2 start with some basic results. in section 3, we establish necessary and sucient conditions for the existence of the -hermitian solution to system (7), and establish an expression of this -hermitian solution when the solvability conditions are satised. in section 4, as an application of theory developed in section 3, we give the solvability conditions for the existence of the -bihermitian to system(8) and give an expression of an -bihermitian solution to (8) when it has this kind solution. we also present a criterion for a quaternion matrix to be -bihermitian. in section 5, we give an algorithm and a numerical example to illustrate the theory developed in this paper.1 preliminariesin the investigation of properties of -hermitian matrices and quaternion matrix involu- tions, a seminal paper was given by horn and zhang 2. they proposed the factorization of an -hermitian matrix a, and considered some properties of quaternion matrix involution- s. recently, he and wang 18 studied some properties of moore-penrose inverse concerning quaternion matrix involutions. a group of formulas for quaternion matrix involutions given in the following lemma.lemma 1. 18 let a hmn be given. then(1) (a ) = (a) , (a) = (a).(2) r(a) = r(a) = r(a ) = r(a a) = r(aa ).(3) (aa) = a(a) = (aa) = (a) a . (4) (aa) = (a)a = (aa) = a (a) . (5) (la) = (la) = (la) = la = ra . (6) (ra) = (ra) = (ra) = ra = la .the following result can be found in 24.lemma 2. let a, b, and c be arbitrary matrices over h with appropriate dimensions. then the fol lowing equalities hold:la(bla) = (bla),(rac )ra = (rac ).the next lemma for ranks of matrices is due to marsaglia and styan 50, which can be generalized to h.lemma 3. let a hmn, b hmk , c hln, d hmp, q hm1 k , and p hln1 be given. then(1) r(a) + r(rab) = r(b) + r(rb a) = r a b .a(2) r(a) + r(c la) = r(c ) + r(alc ) = r .ca b(3) r(b) + r(c ) + r(rb alc ) = r .c 0(4) r(p ) + r(q) + r a blqa b 0 = r c 0p .rp c00q 0in order to establish the -hermitian solution to system (7), we need some results on-hermitian solutions of some matrix equations.lemma 4. let a hmn and b hmn be given. then the matrix equation (3) has an -hermitian solution if and only if rab = 0 and ab = ba. in this case, the general -hermitian solution to (3) can be expressed asx = ab + b(a) aab(a) + lav (la) , (9)where v = v is an arbitrary matrix over h with appropriate size.proof. if equation (3) has an -hermitian solution, then we have rab = raax = 0 and ab = ax a = ba. conversely, if equation (3) has an -hermitian solution, i.e., rab = 0 and ab = ba hold. check thataab + ab(a) aaab(a) = b.now we show that its -hermitian solution can be expressed as (9). assume that x0 is an arbitrary -hermitian solution to (3). set v = x0 . then (9) becomex = ab + b(a) aab(a) + lax0 (la)= ab + b(a) aab(a) + (x0 ab)(i aa)= x0 .this implies that any -hermitian solution of (3) can be represented by (9). thus (9) is the general -hermitian solution to (3).applying lemma 4, we can obtain the following lemma which plays an important role for the development in this paper.lemma 5. let a1 hm1 n, c1 hm1 n, b1 hnm2 , and d1 hnm2 be given. set a1 be =1, f = c1 .d1then system (4) has an -hermitian solution if and only if re f = 0 and ef = f e. in this case, the general -hermitian solution to (4) can be expressed asx = ef + f (e) eef (e) + le u (le ) ,where u = u is an arbitrary matrix over h with appropriate size.proof. note that system (4) has an -hermitian solution if and only if the equation ex = f has an -hermitian solution. we can derive the solvability condition and the expression of the -hermitian solution by lemma 4.2 the solvability conditions and the -hermitian solution to (7)in this section, we give some solvability conditions for (7) to possess a pair of -hermitian solution and to provide an expression of this -hermitian solution when the solvability condi- tions are met. we have the following.theorem 1. let a1 hm1 n, b1 hnm2 , c1 hm1 n, d1 hnm2 , a2 hm3 n1 , b2 3hn1 m4 , c2 hm3 n1 , d2 hn1 m4 , c3 hpn, d3 hpn1 , and a3 = a hpp( i, j, k) be given. pute = a1 b1, f = c1 d1, g = a2 2, h =b c2 d2, (10)c = c3 le , d = d3 lg, m = rc d, s = dlm , (11)a = a3 c3 ef c c3 f (e)c d3 gh d d3 h (g)d3 3 3 3+ c3 eef (e)c + d3 ggh (g)d.(12)3 3then the fol lowing statements are equivalent:(1) system (7) has a pair of -hermitian solutions x and y . (2)ef = f e, gh = h g, re f = 0, rgh = 0, (13)rm rc a = 0, rc a(rd ) = 0. (14)(3)ef = f e, gh = h g, re f = 0, rgh = 0,m m rc a = rc a = rc ad (d) .(4)ef = f e,gh = h g, a1 c1 r a1 = r , r a2 c2 a2 = r, (15)bbbb1 d1 12 d2 2a3 d3 c3 d3 c3 3 a2 0 c2 d a2 0 r d2 d3 b2 0 = r b2 0 , (16) c1 c 0a1 3 0a1d 1 c3 0b10b12a3 c3 d3 c d3 d2 d c3 d3r 3 0a2 b2 = r a + r a . (17) c c 1 2 1 3 a1 00bbd 1 21 c3 b1 00in this case, the -hermitian solution to system (7) can be expressed asx = ef + f (e) eef (e) + le c a(c )(le ) le c sw2 (le c s)1 1 2 le c dm ai + (sd)(le c ) 2 le c (i + sd )a(le c dm )1+ le lc v (le ) + le v1 (le lc ),(18)y = gh + h (g) ggh (g) +11lgm a(d)i + (ss)(lg)2+ lg(i + ss)da(lgm ) + lglm w2 (lglm ) + lglm ls w1 (lg)2+ lgw (lglm ls ) + lgld v (lg) + lgv2 (lgld ),(19)1 22where