The Estimates of Diagonally Dominant Degree and Eigenvalue Inclusion Regions for the Schur Complement of Matrices

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1 dvces i Pure Mhemics Pubished Oie ugus 05 i SciRes hp://wwwscirporg/jour/pm hp://dxdoiorg/0436/pm he Esimes of Digoy Domi Degree d Eigevue Icusio Regios for he Schur Compeme of Mrices Doge Go Deprme of Mhemics Heze Uersiy Shdog Chi Emi: izi_004@6com Receed 9 Juy 05; cceped 3 ugus 05; pubished 6 ugus 05 Copyrigh 05 by uhor d Scieific Reserch Pubishig Ic his wor is icesed uder he Cree Commos ribuio Ierio Licese (CC BY hp://creecommosorg/iceses/by/40/ bsrc he heory of Schur compeme pys impor roe i my fieds such s mrix heory coro heory d compuio mhemics I his pper some ew esimes of digoy -digoy d produc -digoy domi degree o he Schur compeme of mrices re obied which improve some ree resus s ppicio we prese sever ew eigevue icusio regios for he Schur compeme of mrices Fiy we ge umeric exmpe o iusre he dvges of our dered resus Keywords Schur Compeme Gerschgori heorem Digoy Domi Degree Eigevue Iroducio Le deoe he se of N wrie R = C = i N compex mrices { } i ij i j i j i = d = ( ( = { > } = { > ( } N i R i N N i C i N r ii i c ii i We ow h is ced sricy digoy domi mrix if > R i N is ced Osrowsi mrix (see [] if ii i How o cie his pper: Go DJ (05 he Esimes of Digoy Domi Degree d Eigevue Icusio Regios for he Schur Compeme of Mrices dvces i Pure Mhemics hp://dxdoiorg/0436/pm ij We

2 D J Go ( > R R i j N i j ii jj i j SD d OS wi be used o deoe he ses of sricy digoy domi mrices d he ses Osrowsi mrices respecey s show i [] for i d [ 0] we c ii Ri ( ii Ri ( Ci ( d ii Ri Ci he i-h digoy -digoy d produc -digoy domi degree of respecey For β N deoe by β he crdiiy of β d β = N β If βγ N he ( βγ is he submrix of wih row idices i β d coum idices i γ I pricur ( β β is bbrevied o ( β If β is osigur β = β = β β β β β β is ced he Schur compeme of wih respec o ( β he compriso mrix of µ ( = ( ij is defied by ij if i = j ij = ij if i j mrix = ( ij is ced M-mrix if here exis oege mrix B d re umber s > ρ ( B where ρ ( B is he specr rdius of B such h = si B I is ow h is h-mrix if d oy if µ ( is m-mrix d if is m-mrix he he Schur compeme of is so m-mrix d de > 0 (see [3] We deoe by H d M he ses of h-mrices d m-mrices respecey he Schur compeme of mrix is impor pr of mrix heory which hs bee proved o be usefu oos i my fieds such s coro heory sisics d compuio mhemics o of wor hs bee doe o i (see [4]-[8] We ow h he Schur compemes of sricy digoy domi mrices re sricy digoy domi mrices d he Schur compemes of Osrowsi mrices re Osrowsi mrices hese properies hve bee used for derig mrix iequiies i mrix ysis d for he covergece of ierios i umeric ysis (see [9]-[] More impory sudyig he ocios for he eigevues of he Schur compeme is of gre sigificce s show i [] [6] [3]-[8] he pper is orgized s foows I Secio we ge some ew esimes of digoy domi degree o he Schur compeme of mrices I Secio 3 we prese sever ew eigevue icusio regios for he Schur compeme of mrices I Secio 4 we ge umeric exmpe o iusre he dvges of our dered resus he Digoy Domi Degree for he Schur Compeme I his secio we prese sever ew esimes of digoy -digoy d produc -digoy domi degree o he Schur compeme of mrices Lemm [3] If H he µ Lemm [3] If SD or OS he H ie µ ( M Lemm 3 [6] If SD or OS d β N he he Schur compeme of is i SD β or OS β where β = N β is he compeme of β i N d β is he crdiiy of β Lemm 4 [6] Le > b c > b b >0 d 0 he heorem Le s c ( b ( c b + b = β = { i i i } N ( φ { j j j } ij β = he for d r ( β δ j j j j j j j β = < d R R + R ( ( β δ + R + R + R ( j j j j j j j 644

3 D J Go where ( ii ju ii P u = r = mx v v ii ii ii R vu u= v = δ j ( = P rr v Proof Sice β N r ( φ he ( β H d µ ( ( β M hve hus for y ε >0 d we obi For y If R ( β ( µ β β jj jjs s i j i js ( µ ( β jj jjs s s= j j ij ij s = ( ( β ( ( β = R + j ij s i js + ( δ ε ( δ ε ( µ ( β v j j s= From Lemm d Lemm we δ v j + ε ij s = j de s j R j + δ j ε + = de µ ( ( β µ ( ( β i js s= j β deoe B ij v i jv x x > v P ( µ β ii ij s i js 645

4 D J Go he here exiss sufficiey sm posie umber ε 0 such h P x > + ε 0 v ii Cosruc posie digo mrix X = ( B B For p = by (3 we hve Le = X = ( b pq d for p = 3 + by b pp dig x x x + where if = x = Pi + ε 0 i i if = P b pp Rp ( B = b b j = x j 0 0 v i + ε > j= ii R p P ii ( B r v we obi P Pi v = + ε + i p ip ip 0 ip i ε v 0 ip j v v p ii ip i p P i v = P + ε ip 0 ip ip ip ip ip jv v p v p ii 0 ip i p ip + r ip i v ip > v p v p v p ii = ε hus B SD d so + H + Le x be v Sice µ ( ( β j B = µ B M + he B Noe h δ + ε i B he P de B > 0 (4 δ v j + ε v ii de > 0 by (4 we hve ii P i P v i v = + ε > 0 v v v ii P ( ( ii R β > R + δ ε j j j j Le ε 0 he we obi ( Simiry we c prove ( Remr Noe h ( P R v his shows h heorem improves heorem of [7] d [] respecey Nex we prese some ew esimes of -digoy d produc -digoy domi degree of he Schur compeme 0 (3 646

5 = ij β = { i i i } N r N c φ β = he for 0 heorem Le s d d where for y v ( ( { j j j } ( D J Go β = < R β C β R δ C δ (5 j j j j ( ( ( + R β C β + R δ C δ (6 j j j j P = δ η iωω i iωjv ii v = = mx v ω ii i i i i R ωω ωv i ω v ω Q = δ ξ iωω i v ω ii v = i mx v j = ω ii i i ii C ωω v ω i ω v ω Pi = ηr v i Q v i = ξc v i v Proof By Lemm d Lemm we hve µ ( ( β ( β we hve Le ( ( β ( ( β R C ij jsj s s s i j hus for 0 ij ij s = j ( ( j β j j + s β s i j i j s + ( ( β ij ij s j ( ( ( ( j µ β jj s µ β + s i j i j s ij j ( ( sj + s µ β s s i j Simir s he proof of heorem we c prove ij ζ = ( ( µ β i j 647

6 D J Go Simiry we hve By Lemm 4 we hve ij s j ( ( j + s µ β R j δ ζ s i j s ij j ( ( sj + s µ β C s j δ ζ s i j ( ( β ( ( β ( R C Hece (5 hods Simiry we c prove (6 Remr Noe h ( ( ζ R δ ζ C δ ζ j j j j j j ζ R j δ Cj δ ζ = R C ( δ ( δ j j j j ( P R Q C his shows h heorem 3 improves heorem 4 of [] Simir s he proof of heorem we c prove he foowig heorem immediey which improves heorem of [] heorem 3 Le = ( ij β = { i i i } N r N c φ β = { j j j } < d β = ( s he for 0 R ( β ( C ( β R C + δ + δ d R C j j j j j j j j ( ( ( ( ( + R β + C β + R + C δ δ j j j j + R + C j j j j 3 Eigevue Icusio Regios of he Schur Compeme I his secio bsed o hese dered resus i Secio we prese ew eigevue icusio regios for he Schur compeme of mrices heorem 4 Le = ( ij β = { i i i } N r ( φ β = { j j j } < d β = d λ be eigevue of β he here exiss such h s δ λ R R (7 j j j j j Proof By Gerschgori Circe heorem we ow h here exiss hus by Lemm d Lemm we hve such h λ R ( β 648

7 D J Go ie hus (7 hods 0 λ R ( β ij ij s = λ + ( ( β ( ( β jj jjs s = i j i js ij s λ j ( ( j j j s µ β s= s= i js ij s = λ j ( j R ( j + + δ v j δ j µ β s= i js λ R + δ jj j j ij Lemm 5 [] Le such h δ ( λ R R j j j j j = d 0 he for y eigevue µ of here exiss ( ( µ R C heorem 5 Le = ( ij β = { i i i } N r N c φ β = j j j < β = d λ be eigevue of β he for y 0 here exiss such h s ( { } ( ( λ R δ C δ R C (8 j j j j j j Proof By Lemm 5 we ow h here exiss such h herefore ( ( ( ( 0 λ R β C β ( ( λ R β C β ij ij s = λ j ( ( j β jj + s β s= i j i j s ij j ( sj + s β s s= i j ij ij s λ j ( ( j β jj + s β s= i j i j s j ( sj + s β s s= ij i j 649

8 D J Go Simir s he proof of heorem we c prove hus we hve Furher we obi (8 4 Numeric Exmpe ( ( β ij i j + j ( j + s β s= ij s i js ij j ( sj + s β s s= i j ( R δ C ( δ j j ( ( ( ( ( 0 λ R β C β λ R δ C δ j j j j I his secio we prese umeric exmpe o iusre he dvges of our dered resus Exmpe Le By ccuio wih Mb 7 we hve h = 3 7 β = { 3 } ( R = ; R = 9; R = 8; R = ; R = ; ( C = 3; C = 9; C = 4; C = 0; C = 7; δ = 800; δ = 43600; δ = 4500; δ = 4550; δ = 08404; δ = Sice β N r ( by heorem 4 he eigevue icusio se of β is Γ = { λ λ } { λ λ } { λ λ } From heorem 4 of [] he eigevue icusio se of β is { λ λ } { λ λ } { λ λ } Γ = We use Figure o iusre Γ Γ d he eigevues of β re deoed by + i Figure he bue doed ie d gree dshed ie deoe he correspodig discs Γ d Γ respecey Mewhie sice β Nr Nc( by ig = 05 i heorem 5 he eigevue icusio se of β is 650

9 D J Go Figure he bue doed ie d gree dshed ie deoe he correspodig discs Γ d Γ respecey Figure he bue doed ie d gree dshed ie deoe he correspodig discs Γ d Γ respecey { λ λ } { λ λ } { λ λ } Γ = From heorem 5 of [] he eigevue icusio se of β is { λ λ } { λ λ } { λ λ } Γ = We use Figure o iusre Γ Γ d he eigevues of β re deoed by + i Figure he bue doed ie d gree dshed ie deoe he correspodig discs Γ d Γ respecey I is cer h Γ Γ d Γ Γ Refereces [] Cveović Lj (009 New Subcss of h-mrices ppied Mhemics d Compuio hp://dxdoiorg/006/jmc [] Liu JZ d Hug ZJ (00 he Domi Degree d Disc heorem for he Schur Compeme ppied Mhemics d Compuio hp://dxdoiorg/006/jmc [3] Hor R d Johso CR (99 opics i Mrix ysis Cmbridge Uersiy Press New Yor 65

10 D J Go hp://dxdoiorg/007/cbo [4] Crso D d Mrhm (979 Schur Compemes o Digoy Domi Mrices Czechosov Mhemic Jour [5] Irmov KD (989 Ivrice of he Bruer Digo Domice i Gussi Eimiio Moscow Uersiy Compuio Mhemics d Cybereics 9-94 [6] Li B d ssomeros M (997 Douby Digoy Domi Mrices Lier gebr d Is ppicios 6-35 hp://dxdoiorg/006/s ( [7] Smih R (99 Some Iercig Properies of he Schur Compeme of Hermii Mrix Lier gebr d Is ppicios hp://dxdoiorg/006/ (9903-z [8] Zhg FZ (005 he Schur Compeme d Is ppicios Spriger-Verg New Yor hp://dxdoiorg/0007/b05056 [9] Demme JW (997 ppied Numeric Lier gebr SIM Phidephi [0] Goub GH d V Lo CF (996 Mrix Compuios 3rd Ediio Johs Hopis Uersiy Press Bimore [] Kress R (998 Numeric ysis Spriger New Yor hp://dxdoiorg/0007/ [] Xig SH d Zhg SL (006 Covergece ysis of Boc cceered Over-Rexio Iere Mehods for We Boc H-Mrices o Priio π Lier gebr d Is ppicios hp://dxdoiorg/006/j [3] Liu JZ Li JC Hug ZH d Kog X (008 Some Properies o Schur Compeme d Digo Schur Compeme of Some Digoy Domi Mrices Lier gebr d Is ppicios hp://dxdoiorg/006/j [4] Liu JZ d Hug YQ (004 he Schur Compemes of Geerized Douby Digoy Domi Mrices Lier gebr d Is ppicios hp://dxdoiorg/006/j [5] Liu JZ d Hug YQ (004 Some Properies o Schur Compemes of H-Mrices d Digoy Domi Mrices Lier gebr d Is ppicios hp://dxdoiorg/006/j [6] Liu JZ d Hug ZJ (00 he Schur Compemes of γ-digoy d Produc γ-digoy Domi Mrix d heir Disc Seprio Lier gebr d Is ppicios hp://dxdoiorg/006/j00900 [7] Liu JZ d Zhg FZ (005 Disc Seprio of he Schur Compemes of Digoy Domi Mrices d Deermi Bouds SIM Jour o Mrix ysis d ppicios hp://dxdoiorg/037/ [8] Li Y Ouyg SP Co SJ d Wg RW (00 O Digo-Schur Compemes of Boc Digoy Domi Mrices ppied Mhemics d Compuio hp://dxdoiorg/006/jmc