Perron Complements of Strictly Generalized Doubly Diagonally Dominant Matrices

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1 ISSN , Egd, UK Jour of Iformio d Compuig Sciece Vo 5, No 4, 00, pp 6-70 Perro Compemes of Sricy Geerized Douby Digoy Dom Mrices Li Zeg,, Mig Xio d Tig-Zhu Hug Coege of Compuer Sciece d Techoogy, Souhwes Uiersy for Nioies, Chegdu, Sichu, 6004,Chi Schoo of Appied Mhemics, Uiersy of Eecroic Sciece d Techoogy of Chi, Chegdu, Sichu, 60054, Chi (Receied Juy 5, 00, cceped Sepember 9, 00) Absrc As is ow, Meyer roduced he cocep of he Perro compemes of oegie irreducibe mrices I ddio, he Schur compemes of geerized douby digoy dom mrices were roduced by Liu e [Lier Agebr App, 378(004): 3-44] I his pper, properies of he Perro compeme of sricy geerized douby digoy dome mrices re preseed Keywords: geerized douby digoy dom mrix; digoy dom mrix; oegie irreducibe mrix; Perro compeme; Schur compeme Iroducio Le A be mrix, d rec h A is (row) digoy dom if ji, i,,, A is furher sid o be sricy digoy dom if he iequies of () re sric A douby digoy dom mrix (see, eg, [6]) is mrix such h for i j, () jj i d h A is sricy douby digoy dom if he iequies of () re sric We c A geerized douby digoy dom mrix (see, eg, [6]) if here exis proper subse, of :,,, such h, d for i, j where j ( i jj j i j j )( ) (3) i, i i i We c A sricy geerized douby digoy dom mrix if he iequies of (3) re sric Assumig h he mrix order is, we use he sme oio s i pper [6]: D for digoy dom mrices; SD for sricy digoy dom mrices; DD for douby digoy dom mrices; for sricy douby digoy dom mrices;, SDD GDD for geerized douby digoy dom mrices; SGDD, for sricy geerized douby digoy dom () Correspodig uhor Te: E-mi ddress: zeg@6com Pubished by Word Acdemic Press, Word Acdemic Uio

2 6 Le A Z R, : 0, i Li Zeg, e : Perro Compemes of Sricy Geerized Douby Digoy Dom Mrices j If A I B, B 0, B, he A is ced M-mrix The bsoue mrix of A is defied by A The compriso mrix A is defied by A is ced H-mrix if M d Le H, respeciey, i, i j, j A is M-mrix I he foowig we deoe M-mrices d H-mrices by bo A,, be oempy ordered subses of, h cosisig of sricy icresig egers By we sh deoe he submrix of A yig i rows idexed by ddio,, he he pricip submrix, Suppose h A is bbreed o d coums idexed by A If A is osigur, he he Schur compeme of A i, If, i A is gie by S A A A A A A,, (4) where \ Furhermore, he Schur compemes he bee we sudied for rious csses of mrices, icudig: posie defie mrices, M-mrices, ierse M-mrices d oy oegie mrices A we-ow resu due o Crso d Mrhm [] ses h he Schur compemes of sricy digoy dom mrices re digoy dom Ad he Schur compeme of geerized douby digoy dom mrix is geerized douby digoy dom mrix (see, [6]) A remrbe Schur formu is ([8]) de A de( S( A/ A( ))) de A( ) (5) I coecio wh diide d coquer gorhm for compuig he siory disribuio ecor for Mro chi, Meyer [,3] roduced, for oegie d irreducibe mrix A, he oio of he Perro compeme Agi, Le d \ A i A is gie by where The he Perro compeme of, deoes he specr rdius of mrix Rec h s A is irreducibe, A A AI A P A A A A A I A A, (6) he expressio o he righ-hd side of (6) is we defied, d we obsere h mrix d hus ( A I A) 0 P A A, such s P A A is so oegie d irreducibe, d PA A A, so h is M- Meyer [, 3] hs deried seer eresig d usefu properies of Ad such mrices rise i riey of ppicios [9], he bee sudied mos of he 0h ceury, d he receied icresig eio of e (see [4,5]) For geer irreducibe oegie mrices, Johso d Xeophoos [9] iesige whe he Perro compemes re primie or jus irreducibe d hus swer some issues which were rised by Meyer i his erier pper Some of he resus i [, 3, 4, 5] moied his sudy o Perro compemes of sricy geerized douby digoy dom mrices I fc, gie mrix fmiy, is wys eresig o ow wheher some impor properies or srucures of he fmiy of he mrices re ihered by heir submrices or by he mrices ssoced wh he origi mrices mrix I ddio, for y d for y A, e he exeded Perro compeme be he, P A A A A I A A, (7) JIC emi for coribuio: edor@jicorgu

3 Jour of Iformio d Compuig Sciece, Vo 5 (00) No 4, pp which is so we defied sice A A I his pper, we sh show, i Secio, h some Perro compemes of sricy geerized douby digoy dom d oegie irreducibe mrices A re sricy digoy dom d sricy geerized douby digoy dom oy if he Perro roo of A sisfies some codios Perro compemes of sricy geerized douby digoy dom mrices Sice he Schur compemes of sricy geerized douby digoy dom mrices re sricy geerized douby digoy dom, seems ur o s: Whe is he Perro compeme of sricy geerized douby digoy dom mrix sricy geerized douby digoy dom? The, we cosider properies of he Perro compemes of sricy geerized douby digoy dom d oegie irreducibe mrices Lemm ([]) Le A be oegie irreducibe mrix wh specr rdius A, d e d \ The he Perro compeme, is so oegie irreducibe mrix wh specr rdius A P A A A A A I A A,, Lemm ([6]) Le A SGDD, The { i û } or { i û } Lemm 3 ([7]) Le A SD, SDD, or SGDD The A M ; ie, A H Lemm 4 ([5]) Le A be y irreducibe oegie mrix, d fix y oempy se The for y,, wh d, we he for y A, i / / P A P P A /, i We re ow i posio o se he mi resus for Perro compemes of sricy geerized douby digoy dom mrices Theorem 5 Le A be y oegie irreducibe mrix wh specr rdius A d, A SGDD The, for A, if i he P A A is sricy digoy dom d oegie irreducibe mrix Proof Le i, i,, i d j, j,, j, where By Lemm, whou oss of geery, we ssume { i û } (8) d e i A, i d by (8), we ge By i Thus, A 0 i (9) A i JIC emi for subscripio: pubishig@wauorgu

4 64 or Li Zeg, e : Perro Compemes of Sricy Geerized Douby Digoy Dom Mrices Sice A is irreducibe d oegie, we obi So, Thus, Deoe Le x mx{ x, x,, x }, where i 0 i mx A i i mx mx i i s By (9), we he A i x A I A s s s s AI Ax i js s x is he i h compoe of x We he A x i js ii i ( A ) x ( A ) x (0) s () i js i j i j x mx A A i Noe h A is irreducibe d oegie mrix, he A A, x () JIC emi for coribuio: edor@jicorgu

5 Jour of Iformio d Compuig Sciece, Vo 5 (00) No 4, pp so h AI A is M-mrix The we he ( AI A ) 0d, Sice A SGDD, we he, by (3), for y i, j, By (9), we ge So, for j, Deoe he j j j js s s 0 (3) jj j j i j ( s, ) -ery of PA ( / A ( )) by jj ji ji A I A i j jj j j j i jj j mx j i j i j js (4) The, for,,,, we he s s s s i js,, jj ji,, ji AI A j,, j ji ji A I A s i j s j,, j s ji ji A I A s s s jj,, (by(3)) jj s ji ji A I A s s i js s i js JIC emi for subscripio: pubishig@wauorgu

6 66 Li Zeg, e : Perro Compemes of Sricy Geerized Douby Digoy Dom Mrices i j mx i A j,, ( (),()) j j j s ji ji by s s i j mx i A mx ( by(3)) jj i j jj s i s s s A ji s i j jj j jj mx s i s s s ji ( s (0)) by 0( by(4)) I foows h mrix P A A is sricy digoy dom mrix By Lemm, we he h he P A A is oegie irreducibe This compees he proof Remr 6 Accordig o Theorem 5, we c gie simir resu for i We ow gie exmpe o iusre he resu of Theorem 5 Exmpe 7 Le Ad, The, 0 A 3 0, Obiousy, A SGDD where {}, {, 3} d he mrix A is oegie irreducibe 3, A i P A A( ) 4695, is sricy digoy dom From Theorem 5, we he rii resu bou he exeded Perro compemes of sricy geerized douby digoy dom mrices Corory 8 Le A be y oegie irreducibe mrix wh specr rdius A, The, for y A, d A, if i he A SGDD d P A A is sricy digoy dom d oegie irreducibe mrix I Theorem 5, we he bee obied h he Perro compeme of sricy geerized douby digoy dom mrix is sricy digoy dom oy if he codio boe hods Furhermore, we he he foowig heorem i which we show h uder he ssumpios i Theorem 5, he Perro compeme is sricy geerized douby digoy dom JIC emi for coribuio: edor@jicorgu

7 Jour of Iformio d Compuig Sciece, Vo 5 (00) No 4, pp Theorem 9 Le A be y oegie irreducibe mrix wh specr rdius A d, A SGDD The, for A, if i d y proper subse of he P A A is sricy geerized douby digoy dom d oegie irreducibe mrix Proof Le i, i,, i d j, j,, j, where By Lemm, whou oss of geery, we ssume { i û } (5) d e i We show he heorem i wo seps: is sigeo; d or (i) Cosider he cse h cois oy oe eeme Assume If {} i : by Theorem 5, we he i, P A / A( ) SD SGDD If {} i : for y fixed ju d is {} i, e By (4), we he S A A A A i j j s s s s is j, s j u jui ju ju ju j j u j s s ij ss s, i sj s A j j A ji u s j u i j uj u juj A u j A j, Sice A SGDD, we he, by (3), By A j u j u j u j u j j ss s s j uj u j uj, s u j, i d (5), we ge A 0 ji u ji u j i s j j ui ju (6) JIC emi for subscripio: pubishig@wauorgu

8 68 Li Zeg, e : Perro Compemes of Sricy Geerized Douby Digoy Dom Mrices Thus, So, we he (7) A 0 A j u j u j u j u j j, A SGDD, {,}, {3} 3 By Lemm 3, we he A ( A) M Moreoer, s ji u ji u de A 0 (8) Deoe he s, -ery of P A A by The, for i, j /, we he u s s s ju ju ju j is j ji, s u j j s s s ss s A, s A s ij ji u is j ju j A A s s s,, ss s A s s A ji u u ji u j u j u ju j A u u A j j s ji u i s j jui j j A A s ji u u ji u j u j u ju j A u A j s ji u ss s, s A j u j u j u j u j A j s ji u i s j jui j j A A de( S( A / A()) de A (by (5)) de A () JIC emi for coribuio: edor@jicorgu

9 Jour of Iformio d Compuig Sciece, Vo 5 (00) No 4, pp de A A 0( by(7),(8)) So, we obi P( A / A( i )) SGDD, i { i }, () If cois wo eemes Assume {, i i} By Lemm 4, we he PA ( / A( )) PPA ( ( / Ai ( ))/ Ai ( )) SGDD { i, i }, If cois more h wo eemes d, by iducio, we ge P( A / A( )) SGDD Remr 0 Accordig o Theorem 9, we c gie simir resu for i The foowig exmpe c iusre he resu of Theorem 9 Exmpe Le A 4 4, Obiousy, A SGDD 4, where {,}, {3,4} d he mrix A is oegie irreducibe Ad, The, A i 6308, PAA ( ( )) where {, }, is sricy geerized douby digoy dom From Theorem 9, we remr h is esy o erify h he exeded Perro compeme of sricy geerized douby digoy dom mrix is sricy geerized douby digoy dom oy if he codio boe hods Fiy we remr h we my sighy rex he Sric, codio so h our heorems hod for some mrices This is doe by usu ric-coiuy rgume We om furher discussios o his Moreoer, for oegie mrix A, he Perro compeme c be ppied o reserch properies of A uiizig he reio bewee he mrix d s Perro compeme So, gie mrix fmiy, we c cosider furher discussios o Perro compemes of he fmiy of he mrices 3 Refereces [] D Crso, T Mrhm Schur compemes of digoy dom mrices Czech Mh J 979, 9(04): 46-5 [] C D Meyer Ucoupig he Perro eigeecor probem Lier Agebr App 989, 4/5: [3] C D Meyer Sochsic compemeio ucoupig Mro chis d he heory of ery reducibe sysems SIAM Re 989, 3: 40-7 [4] M Neum Ierses of Perro compemes of ierse M-mrices Lier Agebr App 000, 33: 63-7 [5] S M F, M Neum O Perro compemes of oy oegie mrices Lier Agebr App 00, 37: [6] Jizhou Liu, Yuqig Hug, Fuzhe Zhg The Schur compemes of geerized douby digoy dom mrices Lier Agebr App 004, 378: 3-44 [7] Y M Go, X H Wg Creri for geerized digoy dom mrices d M-mrices Lier Agebr App 99, 69: JIC emi for subscripio: pubishig@wauorgu

10 70 Li Zeg, e : Perro Compemes of Sricy Geerized Douby Digoy Dom Mrices [8] F Z Zhg Mrix Theory: Bsic Resus d Techiques New Yor: Spriger, 999 [9] C R Johso, C Xeophoos Irreducibiy d preimiy of Perro compemes: ppicios of he compressed grph, i: Grph Theory d Sprse Mrix Compuio IMA Mh App 993, (56): 0-06 JIC emi for coribuio: edor@jicorgu