On Monotone Eigenvectors of a Max-T Fuzzy Matrix

Transcription

1 Joural of Appled Mathematcs ad hyscs, 08, 6, ISSN Ole: ISSN rt: O Mootoe Egevectors of a Max-T Fuzzy Matrx Qg Wag, Na Q, Zxua Yag, Lfe Su, Lagju eg, Zhudeg Wag * School of Mathematcs ad Statstcs, Yacheg Teachers Uversty, Yacheg, Cha How to cte ths paper: Wag, Q, Q, N, Yag, ZX, Su, LF, eg, LJ ad Wag, Z (08) O Mootoe Egevectors of a Max-T Fuzzy Matrx Joural of Appled Mathematcs ad hyscs, 6, Receved: May 3, 08 Accepted: May 7, 08 ublshed: May 30, 08 Copyrght 08 by authors ad Scetfc Research ublshg Ic Ths work s lcesed uder the Creatve Commos Attrbuto Iteratoal Lcese (CC BY 40) Ope Access Abstract The egevectors of a fuzzy matrx correspod to steady states of a complex dscrete-evets system, characterzed by the gve trasto matrx ad fuzzy state vectors The descrptos of the egespace for matrces the max-łukasewcz algebra, max-m algebra, max-lpotet-m algebra, max-product algebra ad max-drast algebra have bee preseted prevous papers I ths paper, we vestgate the mootoe egevectors a max-t algebra, lst some partcular propertes of the mootoe egevectors max-łukasewcz algebra, max-m algebra, max-lpotet-m algebra, max-product algebra ad max-drast algebra, respectvely, ad llustrate the relatos amog egespaces these algebras by some examples Keywords Fuzzy Matrx, Tragular Norm, Max-T Algebra, Egespace, Mootoe Egevector Itroducto The egeproblem for a fuzzy matrx correspods to fdg a stable state (or all stable states) of the complex dscrete-evets system descrbed by the gve trasto matrx ad fuzzy state vectors Therefore, the vestgato of the egespace structure fuzzy algebras s mportat for applcato Ths problem has bee solved several types of so-called extremal algebras A max-t fuzzy algebra s defed over the terval [ 0, ] ad uses, stead of the covetoal operatos of addto ad multplcato, the operatos of maxmum ad oe of the tragular orms, the so-called t-orm These operatos are exteded a atural way to the Cartesa products of vectors ad matrces The t-orms together wth the t-coorms play a key role fuzzy theory OI: 0436/jamp May 30, Joural of Appled Mathematcs ad hyscs

2 These fuctos have applcatos may areas, such as decso makg, statstcs, game theory, formato ad data fuso, probablty theory, ad rsk maagemet Although there exst varous t-orms ad famles of t-orms (see, eg, []), let us meto the several ma t-orms: the Łukasewcz t-orm, the ödel t-orm, the lpotet mmum t-orm, the product t-orm, ad the drastc t-orm The Łukasewcz t-orm s computed as x y = max x+ y, 0 L The ödel t-orm s the smplest t-orm ad the cojucto s defed as the mmum of the etres, e, x y= m xy, The lpotet mmum t-orm s defed by x 0, f x+ y, y = m { xy, }, otherwse The defto of the product t-orm s x y = x y The drastc tragular t-orm s the weakest orm ad the basc example of a o-dvsble t-orm o ay partally ordered set The drastc tragular t-orm s defed as follows: { xy} { xy} { xy} 0, f max,, x y = m,, f max, = Recetly, avalec et al [] [3] vestgated the steady states of max-łukasewcz fuzzy systems ad mootoe terval egeproblem max-m algebra, Rashd et al [4] dscussed the egespace structure of a max-product fuzzy matrx ad avalec et al [5] studed the egespace structure of a max-drast fuzzy matrx I ths paper, based o these works, we further study egeproblem We vestgate the egevectors a max-t algebra, study mootoe egevectors max-lpotet-m algebra, dscuss the relato betwee the mootoe egevectors max-t algebra ad max-drast algebra, ad llustrate the relatos amog egespaces these algebras by some examples Egevectors a Max-T Algebra Let T be oe of the tragular orms used fuzzy theory, let us deote the real ut terval [ 0, ] by I By the max-t algebra we uderstad the trple ( I,, ) wth the bary operatos = max ad = T o I For gve atural, we wrte N = {,,, } The set of all permutatos o N wll be deoted by The otatos I( ) ad I(, ) deote the set of all vectors ad all square matrces of a gve dmeso over I, respectvely The operatos ad are exteded to matrces ad vectors the stadard way OI: 0436/jamp Joural of Appled Mathematcs ad hyscs

3 max-t algebra cossts for whch A x= x holds true The e- The egeproblem for a gve matrx A I(, ) fdg a egevector x I( ) gespace of A I(, ) s deoted by F = { x I( ) A x= x} Theorem Let,, be three tragular orms o I, A I(, ) ad x I( ) If, x F ad x F, the x F roof If x F ad x F, the Whe, we have that = Thus, x F e, A x x The theorem s proved A x= x, A x= x x= A x Ax A x= x, The vestgato of the egespace structure ca be smplfed by permutg ay vector x I( ) to a o-decreasg form For gve permutatos ϕψ,, we deote by A ϕψ the matrx wth rows permuted by ϕ ad colums permuted by ψ, ad we deote by x ϕ the vector permuted by ϕ Theorem (avalec [6]) Let A I(, ), x I( ) ( ) f ad oly f xϕ F ( Aϕϕ ) We say a vector x I( ) x x F A ad ϕ The, j N, j s creasg f xj holds for ay ad strctly creasg f x xj wheever j The set of all F A ad the set of all F A Smlar o- I I wll be used wthout the codto A x= x creasg egevectors of a matrx A s deoted by ( ) strctly creasg egevectors of a matrx A s deoted by ( ) tato ( ) ad ( ) Theorem 3 Let A I(, ) ad x I ( ) The x F f for every N the followg hold a x x j N, a x = x j N j j j j roof By defto, x F { } s equvalet wth the codto max a x,, a x = x N, f ad oly whch s equvalet to a j x j x for every j N ad aj xj = x for some j N The theorem s proved Theorem 4 Let A I(, ) ad x I ( ) If x F ) a j for all j N, j, ) a x x F A, the t follows from Theorem 3 that roof If ( ) a x x j > j j, the Whe a j =, aj xj = xj > x, ths s a cotradcto Thus, a j for OI: 0436/jamp Joural of Appled Mathematcs ad hyscs

4 every j N, j Notg that s the largest tragular ormo I, we see that x = a x m a, x ad hece a x The theorem s proved I partcular, f x F wth x =, the a = 3 Egevectors Max-Łukasewcz Algebra The followg theorem cotas several logcal cosequeces of the defto of Łukasewcz tragular orm Theorem 3 (Rashd et al [7]) Let abc,, I The ) a L b= b f ad oly f a = or b = 0, ) a L c= b f ad oly f a = + b c or ( a c ad b = 0 ), 3) al c b f ad oly f a + b c, 4) a L c> b f ad oly f a > + b c, 5) f c b, the a L c b Combg Theorem 3 wth Theorem 3, we have the followg theorem Theorem 3 (Rashd et al [7]) Let A I(, ) ad x I ( ) The x F f ad oly f for every N the followg hold: L ) aj + x xj for every j N ad j, ) f =, the x = 0 or aj = + x xj for some j N, 3) f >, the aj = + x xj for some j N The followg theorem descrbes ecessary codtos uder whch a gve square matrx ca have a strctly creasg egevector Theorem 33 (Rashd et al [7]) Let A I(, ) If F Φ, the the L followg codtos are satsfed ) a j for all, j N ad j, ) a = The followg theorem descrbes ecessary ad suffcet codtos uder whch a three-dmesoal fuzzy matrx has a strctly creasg egevector Theorem 34 (Rashd et al [7]) Let A I(, ) The F Φ f ad L oly f the followg codtos are satsfed ) a, a3, a3, for all, j N ad j, ) a =, or a3 a3, 3) a 33 = Example 3 Let us cosder the matrx A = Matrx A satsfes codtos ()-(3) Theorem 34, hece F L Φ ad OI: 0436/jamp Joural of Appled Mathematcs ad hyscs

5 L ( ) {( 0,, 3) 0 0, 3 06} { ( x x x3) x x = x+ x3 = x+ } F A = x x x x = x + 4 Egevectors Max-M Algebra,, 0 03, 0, 07 I the case of the max-m (called also: bottleeck) algebras, the egeproblem has bee studed by may authors ad terestg results descrbg the structure of the egespace have bee foud (see [3] [8] [9] [0] [] []) I partcular, algorthms have bee suggested for computg the maxmal egevector of a gve max-m matrx (see [3]) If the bary operato cocdes wth the mmum operato, the the strctly creasg egespace F creasg egevectors, where the bouds m *, M * I( ) ca be descrbed as a terval of strctly of the terval are defed as follows ( j ) ( j ( ) = ) jk > ( ) = { jk } m A max a k j, M A max a k j, ( ) ( j { ) ( ) } ( ) * * ( j { ) ( ) } m A = max m A j, M A = m M A j If a maxmum of a empty set should be computed the above defto of m A, the we use the fact that max Φ= 0 by usual defto * ( ) The followg theorem has bee proved [6] be a strctly -, e, Theorem 4 (avalec [6]) Let A I(, ) ad x I( ) creasg vector The x F * * f ad oly f m x M * * F = m, M I ( ) Hece, vew of Theorem 4, the structure of F descrbed for ay A I(, ) 5 Egevectors Max-Nlpotet-M Algebra has bee completely We kow that the lpotet mmum orm s left-cotuous ad the R-mplcato geerated from s defed by a b= sup{ x Ia x b} F Moreover, t follows from roposto 5 [4] that ad F form a adjot par, e, they satsfy the followg resdual prcple ad a x b x a b abx,, I, F a b= max x I a x b F If A I(, ), the x F ) for ay j, aj xj x, ) there exst j such that aj xj x For j, s equvalet wth the two codtos: ) f aj x, the aj xj aj x ; ) f aj > x, the aj xj x f ad oly f OI: 0436/jamp Joural of Appled Mathematcs ad hyscs

6 x a x = max a, x, j j F j a) whe j >, xj aj, aj xj = 0, b) whe j =, a x x ad a x = x x a, e, a x a ad hece a > ; 3) f aj = x, the aj xj aj = x ad a x = x x > x x + x > j j j j 6 Egevectors Max-roduct Algebra For every vectors x I( ), defe the quotet vector q q( x) I( ) x, f N \ { }, x+ 0, x+ q =, f N \ { }, x+ = 0, x, f = The, x I ( ) f ad oly f q q( x) = by = fulflls the followg equaltes 0 q, 0 q, 0 q N \, Notg that for ay j, x aj xj x f ad oly f aj x x x + j = x+ x+ xj Thus, t follows from Theorem 3 that Theorem 6 (Rashd et al [4]) Suppose that A I(, ) ad x I ( ) The x F f ad oly f for every N the followg two codtos hold ) aj qq + qj for every j N, j >, ) a = or aj = qq + qj for some j N, j > Whe x F, x 0 > ad t follows from the proof of Theorem 4 that x = a x, e, a = Thus, the followg theorem s a corollary of Theorem 4 Theorem 6 (Rashd et al [4]) If A I(, ) ad F Φ, the the followg codtos satsfed ) a j for all, j N, j, ) a = Ths Theorem descrbes ecessary codtos whch a square matrx ca have a creasg egevector 7 Egevectors Max-rast Algebra For xy, I( ), we have x j OI: 0436/jamp Joural of Appled Mathematcs ad hyscs

7 for every N ( x y) = max { x, y}, { x y} { x y} ( x y) = { x y} m,, f max, =, 0, f max, Theorem 7 (avalec et al [5]) Let A I(, ) ad x I ( ) ( ) the followg codtos hold x F A f ad oly f for every N ) j a for every j N, j >, The ) f x =, the a x, 3) for some j {,, }, aj xj = x Moreover, the followg theorem descrbes ecessary ad suffcet codtos whch a square matrx possesses a strctly creasg egevector Theorem 7 (avalec et al [5]) Let A I(, ) The F oly f the followg codtos are satsfed ) a j for all j N, j >, ) 0 a for all N \ wth a, a a for all k, Nk, wth a, 3) k 4) a = Whe x F, x > 0 ad a, f x =, x = a x = x, f x, a =, 0, otherwse, Φ f ad e, a = Ths shows that codtos () ad (4) Theorem 7 are also straghtforward cosequeces of Theorem 4 The ext theorem characterzes all the egevectors of a gve matrx I other words, the theorem completely descrbes the egespace structure Theorem 73 (avalec et al [5]) Let A I(, ), F ( ) The x F x I f ad oly f the followg codtos are satsfed ) x a Φ, ad = for all N wth a, ) f x =, the x a N \ wth a =, 3) f a, 0 a, the x, 4) f a for some N \, the x = for all 8 The Relatos amog These Egespaces Now we dscuss the relato betwee the mootoe egevectors max-t algebra ad max-drast algebra = If x F Theorem 8 Let A ( a ) I(, ), x I ( ) (,, ) =, the roof Assume that x F j x F Theorem 4 that a j If x =, the ad a = For each N, whe j >, t follows from OI: 0436/jamp Joural of Appled Mathematcs ad hyscs

8 Thus, a x (,, ) Whe a x (,, ) ( ) a x a x = x =,, = =, e, there exst some j {,, } Therefore, x F a x = m, x = x, such that aj xj = x by roposto 33 [5] The theorem s proved Fally, we llustrate the relatos amog egespaces these algebras by two examples Example 8 Let The Thus, x F, but x F (,, ) 3 4 A=, x= A x= =, A x= = 4 3 Ths llustrates that the codto a = = s ecessary Theorem 6 Moreover, by a smple computato, we see that 4 3 Ths example shows that e,,, F eve f F, = I ( 3 ), 3 3 ( ) ( ) F A F A OI: 0436/jamp Joural of Appled Mathematcs ad hyscs

9 Example 8 Let A = The the codtos ()-(4) hold Theorem 34 [5] Thus, F But, e, m *, M * * * * * m =, m = m3 = m4 =, 3 * * * * M = M = M3 = M4 =, = Φ ad F = Φ 9 Coclusos ad Further Works Φ The egeproblem for a fuzzy matrx correspods to fdg a stable state of the complex dscrete-evets system descrbed by the gve trasto matrx ad fuzzy state vectors ad the vestgato of the egespace structure fuzzy algebras s mportat for applcato avalec et al [] [3] have vestgated the steady states of max-łukasewcz fuzzy systems, Rashd et al [4] ad avalec et al [5] have dscussed the egespace structure of a max-product fuzzy matrx ad a max-drast fuzzy matrx, respectvely I ths paper, we vestgated the egevectors a max-t algebra, dscussed mootoe egevectors max-lpotet-m algebra, ad studed the relato betwee the mootoe egevectors max-t algebra ad max-drast algebra I a forthcomg paper, we wll further vestgate mootoe egevectors max-lpotet-m algebra ad max-t algebra Ackowledgemets Ths work s fuded by College Studets ractce Iovato Trag rogram ( Y) Refereces [] Klemet, E, Mesar, R ad ap, E (000) Tragular Norms Treds Logc-Studa Logca Lbrary, Vol 8 Kluwer Academc ublshers, ordrecht [] avalec, M ad Nemcova, Z (07) Steady States of Max-Łukasewcz Fuzzy Systems Fuzzy Sets ad Systems, 35, [3] avalec, M ad lavka, J (00) Mootoe Iterval Egeproblem Max-M Algebra Kyberetka, 46, [4] Rashd, I, avalec, M ad Cmler, R (06) Egespace Structure of a Max-rod Fuzzy Matrx Fuzzy Sets ad Systems, 303, OI: 0436/jamp Joural of Appled Mathematcs ad hyscs

10 [5] avalec, M, Rashd, I ad Cmler, R (04) Egespace Structure of a Max-rast Fuzzy Matrx Fuzzy Sets ad Systems, 49, [6] avalec, M (00) Mootoe Egespace Structure Max-M Algebra Lear Algebra ad Its Applcatos, 345, [7] Rashd, I, avalec, M ad Sergeev, S (0) Egespace of a Three-mesoal Max-Łukasewcz Fuzzy Matrx Kyberetka, 48, [8] Cechlarova, K (99) Egevectors Bottleeck Algebra Lear Algebra ad Its Applcatos, 75, [9] Cughame-ree, RA (99) Mmax Algebra ad Applcatos Fuzzy Sets ad Systems, 4, [0] avalec, M, lavka, J ad oce, (06) Tolerace Types of Iterval Egevectors Max-lus Algebra Iformato Sceces, , [] avalec, M, lavka, J ad Tomaskova, H (04) Iterval Egeproblem Max-M Algebra Lear Algebra ad Its Applcatos, 440, [] avalec, M ad Tomaskova, H (00) Egespace of a Crculat Max-M Matrx Kyberetka, 46, [3] Cechlarova, K (997) Effcet Computato of the reatest Egevector Fuzzy Algebra Tatra Moutas Mathematcal ublcatos,, [4] Baczysk, WM ad Jayaram, B (008) Fuzzy Implcatos Studes Fuzzess ad Soft Computg, Vol 3 Sprger, Chea OI: 0436/jamp Joural of Appled Mathematcs ad hyscs