An Improvement on Disc Separation of the Schur Complement and Bounds for Determinants of Diagonally Dominant Matrices

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1 ISSN , Egd, UK Jor of Iformo d Compg See Vo. 5, No. 3, 2, pp A Improveme o Ds Sepro of he Shr Compeme d Bods for Deerms of Dgoy Dom Mres Zhohog Hg, Tgzh Hg Shoo of Mhem Sees, Uversy of Eero See d Tehoogy of Ch, Chegd, Sh, 654,. R. Ch (Reeved Febrry 2, 29, eped Oober 22, 29) Absr. I hs pper, we mprove he ds sepro of he Shr ompeme of sry dgoy dom mres preseed L [SIAM. J. Mrx A. App., 27 (25): ]. As ppos, we prese some ew bods for deerms of org mres d esmos for egeves of Shr ompeme. By heore yss, we mprove he bods of deerms esbshed Hg [Comp. Mh. App., 5 (25): ]. Keywords: H-mrx; sry (doby) dgoy dom mrx; Shr ompeme; Geršgor s heorem.. Irodo For ozo of egeves d esmos of deerms, my reserhes hve bee proposed, e.g., [-5]. Reey L [6] dsssed he dgoy dom degree of he Shr ompeme of sry dgoy dom mres d preseed he ozo for egeves of he Shr ompeme d some bods for deerms of he sry dgoy dom mres. Hg [7] esmed he bods for deerms of dgoy dom mres, geer H -mres d ero dgoy dom mres. I hs pper, we mprove he dgoy dom degree of he Shr ompeme of dgoy dom mres [6]. Frher, we obew bods for deerms of dgoy dom mres d he esmos of egeves of he Shr ompeme, hese ress mprove he esmos of [6,7]. Le A C be sry dgoy (row) dom mrx (SD ), f d oy f Le If A ( A) ( A) (bbreved ) 2 () C be sry doby dgoy (row) dom mrx (SDD ), f d oy f ( A) ( A) 2 (2) A SDD, b A SD, he, by (2), here exss qe sh h ( ) A (3) m For A ( ) d B ( b ) C, we wre A B, f b for. A re mrx A s ed M -mrx ( M ) f AsI B, sb d s ( B), ( B) s he sper rds of B. Sppose A C, A be ed H -mrx ( H ) f ( A) M, he omprso mrx ( A) ( ) be defed by Correspodg hor. Te.: E-m ddress: zhohoghg@yhoo.. bshed by Word Adem ress, Word Adem Uo

2 Jor of Iformo d Compg See, Vo. 5 (2) No. 3, pp T Le x deoe he rspose of he veor x, d I deoe he dey mrx. Le A C, d N { 2 }. If N, eqs he rdy of. For oempy dex ses N, we deoe by A( ) he sbmrx of A yg he rows ded by d he oms ded by. The sbmrx A( ) be bbreved o A( ). Le N d N, boh rrged resg order. The A A A( ) A( ) A( )[ A( )] A( ) be ed he Shr ompeme wh respe o A( ). Lemm. (See [8]). Le A M. The here exss posve dgo mrx D sh h AD SD. Lemm.2. (See [2]). Le ASDSDD The ( A) M, A H Lemm.3 (See [9]). Le A C, B M. If ( A) B, he A H d B A Remr.. From Lemm.3, we ob mmedey h Lemm.4 (See []). Le Lemm.5 (See []). Le A H A A [ ( )] A SD d m be proper sbse of. The A C AmSD m. A s H mrx f he foowg eqy be hod N N N 2 (4) N f 2 N N2 esef N ˆ 2 ˆ N N N 2 2 N { N ( A)} N { N ( A)} 2. Ds sepro of he Shr ompemes of SD d SDD I hs seo, by dsssg he rer of Shr ompeme of SD d SDD [6]. H, we mprove he dgoy dom degree of he Lemm 2.. Le A SD (or SDD ), { 2 } be proper sbse of N d N { 2 }. For y, deoe JIC em for sbsrpo: pbshg@wau.org.

3 226 Zhohog Hg, e : A Improveme o Ds Sepro of he Shr Compeme d Bods x B [ A( )] () For A SD, he B f H x (5) v v v f v v v 2 esef ˆ () For A SDD, d be sh s (3), he B f v H x (6) { v } v { } v v f { } v { } { } v esef ˆ roof. Cosder he foowg wo ses: (): N { } N2 ; (): N { } N2 { } Aordg o Lemm.5, we ob eqes (5) d (6). Frher, by Lemm.2, B ( B ) M he de B. The eqy se foows from oy rgme (wh x B d eg ). L [6] defed he foowg : I hs pper, for he smpy, we e () If A SD, he ( A) vv v m v vv (7) JIC em for orbo: edor@.org.

4 Jor of Iformo d Compg See, Vo. 5 (2) No. 3, pp (b) If ˆ A SDD, b A SD, he v v v v (8) v v { } v v { } (9) For he oveee of omprso, we gve some ress of [6]: Theorem 2. [6, Theorem ]. Le A SD, { 2 } N N { 2 } s A ( r ) be defed s (7). The d Corory 2. [6, Corory ]. Le ( A) ( A) ( A) ( A) ( A) ( A) A SD d e { 2 }. The ( A) ( A) ( A) A ( A) Theorem 2.2 [6, Theorem 2]. Le A SDD, d be sh s (3). The for y dex se og, wrg { 2 } N { 2 }, d A ( r ). The d ( ) A ( A) ( ) A v v ( A) ( ) A ( ) A ( A) ( ) A v v ( A) ( ) A I hs pper, we repe [6] by ˆ d by he smr wy o he proof of Theorem d 2 [6], he we ob he smr ress s Theorem 2., 2.2 d Corory 2.. Theorem 2.3. Le A SD, { 2 } N N { 2 } s ˆ be defed s (8), ( ). The d A r Corory 2.2. Le Theorem 2.4. Le ( A) ( A) ( A) ˆ ( A) ( A) ( A) ˆ A SD d e { 2 }. The ( A) ( A) ( A) A ( A) A SDD, d be sh s (3). The for y dex se og, JIC em for sbsrpo: pbshg@wau.org.

5 228 Zhohog Hg, e : A Improveme o Ds Sepro of he Shr Compeme d Bods A s wrg { 2 } N { 2 }, s defed s (9) d ( ). The ( A) ( ) ( ) A A d roof. Se ( A) ( A) ( A) A SDD, by (9), we hve v v { } { } v v { } v v { } v v { } Frher, ordg o Lemm 2., by he smr wy o he proof of Theorem 2 [6], we ompee he proof of Theorem 2.4. Remr 2.. By omprso, we ob h ˆ d Ths, we mprove Theorem, 2 d Corory 2 [6]. 3. Bods for deerms of SD d SDD For he oveee of omprso, we se he sme deoes s [6]. Le {J,J2,,J }be rerrgeme of he eemes N {2 } We Deoe { } 2 { } s { } N. The wh { },, 2,,, d [ A( )] Le represe y rerrgeme { 2 } of he eemes N wh 2. [ ( )] A [ ( )] I hs seo, we se d { } A o repe d [6, Theorem 3 d Theorem 4], respevey. The we ob he foowg ress. Theorem 3.. Le A SD. The Remr 3.. Se, he [ A( )] A A [ ( )] de [ A( )] m [ ( )] A [ A( )] [ A( )] Ths, bods for deerms Theorem 3. re beer h h of [6, Theorem 3]. Espey, we ssme {} 2 { } 2 { 2 }, s {2 } N The {} 2 wh d [ A( )] JIC em for orbo: edor@.org.

6 Jor of Iformo d Compg See, Vo. 5 (2) No. 3, pp [ A( )] v v v v The, we ob he foowg Theorem. v Theorem 3.2. Le A SD d s defed (). The v () v v v v v de A v v v Obvosy, we mprove he foowg Theorem 3.3 [7, Theorem ]. Theorem 3.3 [7, Theorem ]. Le A SD. The m dea m v v v v m For ogos res of SDD, e deoe rerrgemes of he eemes N wh { }, whe be sh s (3). Theorem 3.4. Le A SDD, d A SD d wh be sh s (3). The v [ A( )] v [ ( )] de A A v{ } vv [ A( )] v m [ ( )] A v { } vv Remr 3.2. Se, he [ A( )] v v { } Ths, we ob he beer bods for deerms h he bods [6, Theorem 4]. Theorem 3.5. Le A H. The v vv m { R } dea { R } [( AX)( )] R [( )( )] AX x d X dg( x x2x ) be posve dgo mrx d ssfes AX SD. JIC em for sbsrpo: pbshg@wau.org.

7 23 Zhohog Hg, e : A Improveme o Ds Sepro of he Shr Compeme d Bods roof. Se A H, by Lemm., he, here exss posve dgo mrx X ssfy SD Frher, ordg o Theorem 3., we ob he ress. Corory 3.. Le A C d ssfes (4). The xvv dea xvv x v x v ( A) N 2 x N N2 N2 m N N2, N N N N 2 AX. roof. Aordg o Lemm.5, we see he posve dgo mrx X d s eemes sh s (), he AX SD. Frher, by Theorem 3.5, we ompee he proof of Corory 3.. () 4. Bods for he Shr ompeme of SD d SDD I hs seo, ordg o Geshgor s heorem, we gve he ozo for egeves of he Shr ompeme of SD d SDD. Frher, we mprove he ower bod for egeves of Shr ompeme of SD [6,Theorem 5]. Theorem 4.. Le A SD, ˆ be defed s (7) d, be defed s Lemm 2., ( A ) deoe he se of egeves of A, d A ( r ). The, for y egeve of he Shr ompeme of SD, we hve. m[ ( A) ˆ ] [ ( A) ˆ ] roof. By Geshgor s heorem, we ob h ( A) Ths ( A) ( ) A Frher, ordg o Theorem 2., we hve ( ) ˆ ( ) ˆ A A Ths, we ompee he proof of Theorem 4.. Remr 4.. By omprso, we ow h he bove bods for egeves re more re h he bods [6, Theorem 5]. Theorem 4.2. Le A SDD d A SD, wh be sh s (3), be defed s (3),, be defed s Lemm 2., ( A ) deoe he se of egeves of A, d deoe A ( r ). The, for y egeve of he Shr ompeme of SD, we hve. m ( ) ( ) A A roof. Aordg o Theorem 3.2, by he smr wy o he proof of Theorem 4., we ob Theorem Exmpes I hs seo, we prese some exmpes o sre hese bods hs pper re more effey JIC em for orbo: edor@.org.

8 Jor of Iformo d Compg See, Vo. 5 (2) No. 3, pp h bods [6,7]. Exmpe. Le 3 A 2 4 dea27 2 By Theorem 3.: de A 3549 By [6, Theorem 3]: 975 de A 4275 By Theorem 3.3 ([7, Theorem ]): 56 dea Exmpe 2. Le 3 A 2 de A By Theorem 3.2: 9de A 39. By Theorem 3 of [7]: 6 de A 53 Exmpe 3. Le A Obvosy, A be sry dgoy dom mrx. Who oss of geery, we ssme { 2 } { 34 }. The A ( A) { } Aordg o Theorem 4., we hve m[ ( A) ˆ ] 24 [ ( A) ˆ ] 6 Aordg o Theorem 5 [7], we hve m[ ( A) ]( 27) By mer omprso, we ow h he ower bod for egeves s more re h he ower bod [7]. Exmpe 4. Le A Obvosy, A SDD b A SD, d be sh s (3). Who oss of geery, we ssme { 2} {3 4}. The A ( A) { } m ( ) 3 22 ( ) 2 7 A A JIC em for sbsrpo: pbshg@wau.org.

9 Aowedgemes Zhohog Hg, e : A Improveme o Ds Sepro of he Shr Compeme d Bods Ths reserh ws sppored by NSFC (69735). 7. Referees [] B. Kr, T. H. e. A deerm ower bod. Ler Agebr App. 2, 326: [2] S. Che. A ower bod for he mmm egeve of he Hdmrd prod of mres. Ler Agebr App. 24, 378: [3] S. M. F, C. R. Johso, R. L. Smh, V.D. Dresshe. egeve oo for oegve d Z -mres. Ler Agebr App. 998, 277: [4] C. K. L, R. C. L. A oe o egeves of perrbed Herm mres. Ler Agebr App. 25, 395: [5] H. Y, Y. Te, H. Ish. Noegve deerm of regr mrx: Is defo d ppos o mvre yss. Ler Agebr App. 26, 47: [6] J. L. Ds sepro of he Shr ompeme of dgoy dom dres d deerm bods. SIAM. J. Mrx A. App. 25, 27: [7] T. Z. Hg. Esmes for er deerms. Comp. Mh. App. 25, 5: [8] A. Berm, R. J. emmos. Noegve Mres he Mhem Sees. New Yor: Adem ress, 979. [9] W. L. O Nersov mres. Ler Agebr App. 998, 28: [] D. Cso, T. Mrhm. Shr ompemes o dgoy dom mres. Czeh. Mh. J. 979, 29: [] G. X. She. Some ew deerme odos for osgr H -mrx. J. Eg. Mh. 998, 4: 2-27 ( Chese). [2] B. L, M. Tssomeros. Doby dgoy dom mres. Ler Agebr App. 997, 26: JIC em for orbo: edor@.org.