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J ou rn al of M a t he m a tica l R esea rch ＆ E x p osit ion V o1． 2 3 ， N o． 1 ，1 5— 2 0 ，F e br ua r y ，20 0 3 S y s t e m s of M a t r ix E q u a t i on s ov e r a C e n t r a l A lg e b r a WA N G Qing— wen， L I S hang— zhi (Dept． of M ath． ， University of Science and T echnology of China， Hefei 230026， C hina) A b st r a c t ：a utonl orphism an d char tf lered ． N ec es sary a n d s uf f iper(skew)selfconjugate solutions of the systems are given， respectively． L et Q b e a f i n it e di m en sion al c en tra l algeb ra w it h all in v olu to ria l an t i— ≠2．T wo sy stems of m a trix equations over Q a re consid． c ien t c on di tion s f or t he ex is t en c es of gen era l so lu t ion s，an d K ey w ords： central algebra； system of matrix equations； per(skew)selfl c0l1 ． ． ugate Ina． trix ；r egu la r m a tr ix qu ad ru p le． C lassif i cation：A MS(2000) 15A 24， 15A 33／ C LC number：O151． 21 D ocum ent code： A A rt icle ID ： 1000— -34 1X (2003)01． ． 0015． ． 06 1 ．I n t r o d u c t i o n T hr o ug h o ut th is p ap er ， w e deno t e a f i n i te dim ensio na l centr al alg ebr a w it h a n inv o lu． tion l1， P． 112， Def i nition 1} over a f i eld F by Q and chart~ ≠ 2 ， the set of au m × 佗 matrices over Q by Q Xn ， the set of all n × n invertible matrices over Q by G L n( Q) ， the set of all m × n matrices over Qf 1 by Q” “ 『 1， and an i × i identity matrix by 厶．Let A = (aij ) ∈Q “ ， A = ( (a竹 1一 j+1，per(skew)selfconjugate if A = A (- A 1． T he set of al l per(skew)selfconj ugate matrices is denoted by C S )． For matrices A ， B over Q， it is easy to veri f y that (A ) = A ， (A B ) = B A ． Def i ne ( )一 = A 一 i f A is invertible． Suppose A ， B ∈Q“ “ ， C ， D ∈Q” is ca lle d a r eg ula r m a tr ix qu a dr up l e if ther e ex i sts in ver tib le ． 一 件1)) ∈Q“ ． T hen A ∈ Q“ “ is called ， then (A ， B ， C ， D ) A B+∈ F su ch tha t A +， CAD a r e M a n y p r ob lem s i n sy st em s a nd con tr ol theor y r e qui re t he solut ion of S y lv ester ’S m a t ri x e qu a tion — X B=C or i ts g ener al iza tion A X— Y B=C． W ．E ．R oth f21 gave ne cessa rm atr i x e qu a tion A X B — C X Ddif f erential equations [3]．y a nd suf f ici ent co n di tion s fo r th e co nsisten cy of the tw o m a tr ix equ at io ns=E ap p ea rs in th e num er ic al solu tion of i m p li ci t or din a r v ．T he R e c e i v e d d a t e ：20 0 0． 0 2 ． 2 5 F oundation item：Supported by the National N atural Science F oundation of C hina (1007107s) and N atural Science Foundation of Shandong P rovince (Q99A 08) B iogr ap hy ： W A NG Qi ng· Well (1964- )， in[ti e， Ph． D ． ， Professor． 一15— 维普资讯 http://www.cqvip.com