Max-norm and Square-max Norm of Fuzzy Matrices

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1 Jourl of themtics d Iformtics Vol 3, 05, 5-40 ISSN: (P), (olie) Published 0 ugust 05 wwwreserchmthsciorg Jourl of x-orm d Squre-mx Norm of Fuzzy trices Sum ity Deprtmet of pplied themtics with Oceology d Compute Progrmmig Vidysgr Uiversity, idpore - 70, Idi emil: mitysum0@gmilcom Received July 05; ccepted 5 ugust 05 bstrct Fuzzy mtrices ply importt role to model severl ucerti systems I this pper, two types of orms, viz mx orm d squre-mx orm of fuzzy mtrices re itroduced lso, severl properties re ivestigted Keywords: x orm, squre-mx orm, properties of mx orm Itroductio he study of lier lgebr hs become more d more populr i the lst few decdes People re ttrcted to this subject becuse of its beuty d its coectio with my other pure d pplied res I theoreticl developmet of the subject s well s i my pplictio, oe ofte eeds to mesure the legth of vectors For this purpose, orm fuctios re cosider o vector spce orm o rel vector spce V is fuctio :V R stisfyig u > 0 for y ozero u V ru r u for y r R d u V 3 u + v u + v for y u, v V he orm is mesure of the size of the vector u where coditio () requires the size to be positive, coditio () requires the size to be scled s the vector is scled, d coditio 3 (3) is kow s the trigle iequlity d hs its origi i the otio of distce i R he coditio () is clled homogeeous coditio d this coditio esure tht the orm of the zero vector i V is 0; this coditio is ofte icluded i the defiitio of orm Commo exmple of orms o R re the l p orms,where p, defied by l p p p ( u) { u j } if < j 5 p d l p (u) mx u if p j j t for y u ( u, u,, u ) R Note tht if oe defie l p fuctio o R s defie bove with 0 < p <, the it does ot stisfy the trigle iequlity, hece is ot orm Give orm o rel vector spce V, oe c compre the orms of vectors,

2 Sum ity discuss covergece of sequece of vectors, study limits d cotiuity of trsformtios, d cosider pproximtio problems such s fidig the erest elemet i subset or subspce of V to give vector hese problems rise turlly i lysis, umericl lysis, differetil equtios, rkov chis, etc he orm of mtrix is mesure of how lrge its elemets re It is wy of determiig the "size" of mtrix tht is ecessrily relted to how my rows or colums the mtrix hs he orm of squre mtrix is o egtive rel umber deoted by here re severl differet wys of defiig mtrix orm but they ll shre the followig properties: 0 for y squre mtrix 0 iff the mtrix 0 3 K K for y scler K 4 + B + B for y squre mtrix, B 5 B B Differet types of mtrix orm: he -orm mx ( ) j (the mximum bsolute colum sum) Simply we sum the bsolute vlues dow ech colum d the tke the biggest swer ( useful remider is tht "" is tll, thi chrcter d colum is tll, thi qutity) he ifiity orm i mx ( ) i he ifiity orm of squre mtrix is the mximum of the bsolute row sum Simply we sum the bsolute vlues log ech row d the tke the biggest swer Euclide orm E j i j he Euclide orm of squre mtrix is the squre root of the sum of ll the squres of the elemets his is similr to ordiry "Pythgore" legth where the size of vector is foud by tkig the squre root of the sum of the squres of ll the elemets y defiitio you c defie of which stisfies the five coditio metioed t the begiig of this sectio is defiitio of orm here re my my possibilities, but the three give bove re mog the most commoly used Like vector orm d mtrix orm, orm of fuzzy mtrix is lso fuctio : ( F) [0,] which stisfies the followig properties ( ) 6

3 x-orm d Squre-mx Norm of Fuzzy trices 0 for y fuzzy mtrix 0 iff the fuzzy mtrix 0 3 K K for y scler K [0,] 4 + B + B for y two fuzzy mtrix d B 5 B B for y two fuzzy mtrix d B I this project pper we will defie differet type of orm o fuzzy mtrices Why study differet orms? Differet orm o vector spce c give rise to differet geometricl d lyticl structures I ifiite dimesiol vector spce, the covergece of sequece c vry depedig o the choice of orm his pheome leds to my iterestig questios d reserch i lysis d fuctiol lysis I fiite dimesiol vector spce V, two orm d equivlet if there exist two positive costt such tht v v for ll v re sid to be First, for give sequece it my be esier to prove covergece with respect to oe orm rther th other I pplictio such s umericl lysis oe would like to use orm tht c determie covergece efficietly herefore, it is good ide to hve kowledge of differet orms Secod, sometimes specific orm my be eeded to del with certi problem For istce, if oe trvels i htt d wts to mesure the distce from loctio mrked s the origi (0,0) to destitio mrked s ( x, y) o the mp, oe my use the l orm of ( x, y), which mesures the stright lie distce betwee two poits, or oe my eed to use the l orm of v, which mesures the distce for txi cb to drive from (0,0) to ( x, y) he l orm is sometimes referred to s the txi cb orm for this reso I pproximtio theory, solutios of problem c vry with differet problems For exmple, if W is subspce of R d v does ot belogs to W, the for < p < there is uique u 0 W such tht v u 0 v u for ll u W, but the uiqueess coditio my fil if p or o see cocrete exmple let v (,0) d W {(0, y) : y R} he for ll y [,] we hve v (0, y) v w for ll w W For some problems, hvig uique pproximtio is good, but for others it my be better to hve my so tht oe of them c be chose to stisfy dditiol coditios 3 Fuzzy mtrix Fuzzy mtrices were itroduce for the first time by homso [4], who discussed the 7

4 Sum ity covergece of powers of fuzzy mtrix Rgb et l [33,34] preseted some properties of the mi-mx compositio of fuzzy mtrices Hshimoto [8,9] studied the coicl form of trsitive fuzzy mtrix Hemshi et l [0] Ivestigted itertes of fuzzy circult mtrices Derermit theory, powers d ilpotet coditios of mtrices over distributive lttice re cosider by Zhg [43] d [4] fter tht Pl, Bhowmik, dk, Shyml, odl hve doe lot of works o fuzzy, ituitioistic fuzzy, itervl-vlued fuzzy, etc mtrices [-,4,5,7-3,35-39] Boole mtrix is mtrix with elemets ech hs vlue 0 or fuzzy mtrix is mtrix with elemets hvig vlues i the closed itervl [0,] We c still see tht ll fuzzy mtrices re mtrices but every mtrix i geerl is ot fuzzy mtrix We see the fuzzy itervl ie, the uit itervl is subset of rels hus mtrix i geerl is ot fuzzy mtrix sice the uit itervl [0,] is cotied i the set of rels he big questio is c we dd two fuzzy mtrices d B d get the sum of them to be fuzzy mtrix he swer i geerl is ot possible for the sum of two fuzzy mtrices my tur out to be mtrix which is ot fuzzy mtrix If we dd bove two fuzzy mtrix d B the ll etries i +B will ot lie i [0,], hece +B is oly just mtrix d ot fuzzy mtrix So oly i cse of fuzzy mtrices the mx or mi opertio re defied Clerly uder the mx or mi opertios the resultt mtrix is gi fuzzy mtrix I geerl to dd two mtrix we use mx opertio Now we wish to fid the product of two fuzzy mtrices X d Y where X d Y re comptible uder multiplictio We see the product of two fuzzy mtrices uder usul mtrix multiplictio is ot fuzzy mtrix So we eed to defie comptible opertio logous to product so tht the product gi hppes to be fuzzy mtrix However eve for this ew opertio if the product XY is to be defied we eed the umber of colums of X is equl to the umber of rows of Y he two types of opertios which we c hve re mx-mi opertio d mi-mx opertio I fuzzy mtrices oly the elemets re ucert while rows d colums re tke s certi But i my rel life situtio we observe tht rows d colums lso be ucerti For exmple, i fuzzy grph the vertices d edges both re ucerti So, if we represet fuzzy grph i mtrix form where the membership vlues of vertices d edges represets the membership vlues of rows d colums d elemets represet the membership vlues of the correspodig edge ht is, i this mtrices rows d colums ll re ucerti We cll this types of mtrices re fuzzy mtrices with fuzzy rows d colums [r (i)][c (j)][ ] m be fuzzy mtrix with fuzzy rows d colums of order m Here, i,,,m; j,,, represets the th elemets of, r (i) d c (j) represets the membership vlues of ith row d jth colum respectively for i,,,m; j,,, Defiitio [4] fuzzy mtrix of order $m\times $ is defied s <, > where is the membership vlue of the -th elemet i fuzzy mtrix R is clled reflexive iff r for ll i,,, It is clled α -reflexive iff r ii α for ll i,,, where α [0,] It is clled wekly reflexive iff r for ll j,,, fuzzy mtrix R is clled irreflexive iff r 0 for ll ii r i,,, ii ii 8

5 x-orm d Squre-mx Norm of Fuzzy trices Defiitio fuzzy mtrix S is clled symmetric iff s s ji for ll j,,, It is clled tisymmetric iff S S I where I is the usul uit mtrix Note tht the coditio S S I, mes tht s 0 for ll i j d s ji s for ll i So if S the s 0, which the crisp cse ii ji Defiitio 3 fuzzy mtrix N is clled ilpotet iff N 0 (the zero mtrix) If m N 0 d N m 0 ; m the N is clled ilpotet of degree m fuzzy mtrix E is clled idempotet iff E E It is clled trsitive iff E E It is clled compct iff E E Defiitio 4 trigulr fuzzy mtrix of order,α, β is the m re left d right spred of m is defied s ( ) m where th elemet of, m is the me vlue of respectively d α, β 4 x orm We lredy kow tht orm of fuzzy mtrix c be defie i severl wys It is lso kow to us tht every orm must be stisfied the five coditio which lredy we discuss i itroductio Now we defie ew type of orm clled mx orm which gives the mximum elemet of the fuzzy mtrix x orm of fuzzy mtrix is deoted by d defied by j Lemm ll the coditio of orm re stisfied by Proof: Let us cosider We lso cosider j So, d d d j B b b b B b b b (i) Clerly 0 d 0 iff j ie iff 0, for ll, ie iff 0 kl kl 0 b b b j b b b 9

6 (ii) Let α [0,] the α if α > the α If If j α j α < the α α the obviously (iii) Now + B if If α α α So, So, Sum ity α α α α α α α α α α α α α bkl the ( ) bkl So, j bkl the ( ) So, j herefore + B + B, for ll, B ( F) + B B + B + B + B (iv) Now ibi ibi ibi i i i ibi ibi ibi B i i i ibi ibi ibi i i i s {, b } d b {, b } therefore mximum elemet of B is less th or equl to miimum of d b kl Here the symbol used to idicte mximum elemet d the symbol used to idicte miimum elemet hus, B B Hece ll the codios of orm re proved 5 Properties of mx orm Properties For every fuzzy mtrix, lwys hold 30

7 x-orm d Squre-mx Norm of Fuzzy trices Proof: s mximum elemet of d trivilly re equl so the bove property hold Properties For y two fuzzy mtrix d B i, ( + + B lwys hold Proof: ( + + B [usig first property of mx-orm] + B [from the defiitio of mx-orm] + B [usig first property of mx-orm] Properties 3 If d B re two fuzzy mtrices d Proof: s B therefore b for ll i, j b B j j Exmple Let d B d B 07 So, B < B the B Properties 4 If d B re two fuzzy mtrices d for ll C B the Proof: s B therefore b for ll i, j his implies tht c b c for ll vlues of ; i ; j So, j c j b c C BC c C BC Properties 5 For y two fuzzy mtrix d B( ) be equl but B B lwys hold B d B my or my ot Exmple Let d B

8 Sum ity B d B B 06 d B 06 So, B B Properties 6 If (i) (ii) the Properties 7 x orm follows Lplce trsform Proof: Let, B ( F) d α, β [0,] the α + βb α + βb α + β B Defiitio 5 Defie mppig d : ( F) ( F) [0,] s for ll, B i d (, + B Propositio he bove mppig d stisfies the followig coditio for ll (i) d (, 0 d d (, 0 iff B 0 (ii) d (, d( ) (iii) d (, d(, + d ( for ll, C i Proof: (i) d (, B 0 [by first coditio of orm] + gi d, 0 + B 0 ( + B 0 0 d B 0 (ii) d(, + B B d( ) (iii), C i + (, B + B + C + B + C + B + C + C + + ( B C + C + B + C d (, + d( d + ( + ) So, d (, d(, + d ( for ll, C i Exmple 3 Let , B d

9 x-orm d Squre-mx Norm of Fuzzy trices C B (i) d, + B 09 > 0 ( (ii) B , the B So, d (, d( ) (iii) Now + C d B + C C 09 d B + C 07 he d (, + d( So, d (, d(, + d( heorem If,, B ( F) the d (, + d( + B ) d (, ) + d( B ) Proof: d (, + d (, B ) + B + + B + B + + B ( + ) + ( B + B ) + + B + B d (, ) + d( B ) heorem If, B ( F) d B the d(, d ( for ll C Proof: s B So, B + C B + + C B + C d(, d( for ll C C Defiitio 6 Defie mppig d : ( F) ( F) [0,] s d (, mi, B } for ll, B i (F ) { Propositio he bove mppig d stisfies the followig coditio for ll i, C 33

10 Sum ity (i) d (, 0 d d (, 0 iff 0 or B 0 or both B 0 (ii) d (, d ( ) Proof: (i) d (, B ) mi {, B } 0 s 0 d B 0 Now d (, mi, B } 0 { 0 or B 0 or both B 0 either 0 or B 0 or both B 0 (ii) d (, mi, B } mi B, } d ( ) { { Propositio 3 If, B ( F) d B the d (, d ( for ll C Proof: Sice B, so B Now d (, mi, C } d d ( mi B, C } { { Cse-: If B C the d (, B d ( ie d (, d ( Cse-: If C B the d (, C d ( Cse-3: If C B the d (, d d C So, d (, d ( herefore d (, d ( for ll C Defiitio 7 For ll i we defie sup { x ( F) : x > } { x ( F) : x < } if equ { x ( F) : x } ( F) sup if equ Clerly ( he set sup is clled mx-superior to, if is clled mx-iferior to d equ is clled mx-equivlet to heorem 3 For ech i the followig results hold true (i) If X sup (or if or equ is the trspose of X (ii) If ) the sup, if d equ X is lso i sup (or if or equ 3 the ) where X 34

11 x-orm d Squre-mx Norm of Fuzzy trices (iii) (iv) 3 equ for ll i Proof: (i) Sice mximum elemet of d re equl therefore So, if X sup the X > X > herefore, (ii) sup if X sup Similrly other two cses lso hold > < equ So, we c write < < (i) From bove reltio it is cler tht mximum elemet of is grter th mximum elemet of d mximum elemet of 3 herefore, mximum elemet of mximum elemet of his implies (iii) We kow tht B B he we hve So, (iv) s therefore equ [usig(i)] for ll i 6 Squre-mx orm Here we will defie other ew orm of fuzzy mtrix med Squre-x orm I this orm t first we will fid the mximum elemet of the fuzzy mtrix d the squre it Squre-mx orm of fuzzy mtrix is deoted by d defie by S ( ) ( ) j Lemm ll the coditios of orm re stisfied by Proof: Let us cosider d b b B b b b b S S ( ) ( ) j b b b 35

12 Let mximum elemet of be d mximum elemet of B be S ( ) d B S (i) Obviously 0 d S Sum ity ie i j kl, b ie b ( b kl ) j 0 S iff 0 (ii) Now we defie sclr multiplictio of mtrix s follows if α > S S α for ll i, j α if α S So, if α > the α α d if herefore S α < the α α α S S S S S 36 kl iff 0 ie iff 0 for ll i, j, ie S S α α for ll α [0,] (ii) Now + B If < bkl the ( ) bkl d the + B B + B S S S S gi if > bkl the ( ) d the + B + B S S S S (iii) ibi ibi ibi i i i Now ibi ibi ibi B i i i ibi ibi ibi i i i s {, b} d b {, b } therefore mximum elemet of B is less th or equl to miimum of d b kl Here the symbol used to idicte mximum elemet d the symbol used to idicte miimum elemet So, B B S S S Hece ll the coditios of orm re stiesfied by Squre-mx orm S

13 x-orm d Squre-mx Norm of Fuzzy trices heorem 4 dditio of two orm of fuzzy mtrices is lso orm Proof: Let d be two orm o Now let us cosider fuctio : ( F) [0,] defie by (i) Sice 0 d 0 so, 0 gi 0 d 0 iff 0 So, 0 iff 0 (ii) Let α F the α α + α α + α [s α ( + ) α (iii) Let, B ( F) + B d be two orm] + B + + B + B + B ( ) + ( B + B + + ) + B So, (iv) + B + B for ll, B ( F) B B + B B + B B + B + B + B ( + )( B + B ) B So, B B for ll, B ( F) 37 + for ll herefore fulfill ll the coditios of orm d hece is orm o heorem 5 Sclr multiplictio of orm of fuzzy mtrices is ot orm Proof : Let be orm o Cse-: Now let : ( F) [0,] be fuctio defie by d C (0,) (i) s 0 d C > 0, so 0 d 0 iff 0 (ii) Let α F the α C α C α [s is orm] α ( C ) α for ll α F (iii) + B C + C + B ) C + C B ( + B for ll, B ( F) (iv) B C B C B C B B C [ Q C (0,)] C for ll

14 Sum ity So, dose ot fulfill ll the codios of orm herefore is ot orm o Cse-: s 0 for ll so, C ever be egtive Cse-3: Let C0, the C 0 lwys hold So, 0 d it does ot imply 0herefore C 0 Cse-4: Let C >, the C does ot belogs to [0,] for ll C > From bove fore cses it is cler tht sclr multiplictio of orm ever become orm 7 Coclusio I this pper, we defie mx-orm d squre-mx orm of fuzzy mtrices I differet situtio we use differet orm Somewhere mx orm is suitble to use th squre-mx orm, somewhere squre-mx orm is suitble th mx orm We lredy prove tht mx orm stisfied Lplce trsformtio So mx orm is very importt thigs i pplictio re Usig these orm we c defie coditiol umber to check whether system of lier equtio is ill posed or well posed Norm of fuzzy mtrices c tke effective cotributio to solve fuzzy system of lier equtio Similrly we c defie orm o trigulr fuzzy mtrix, circult trigulr fuzzy mtrix, fuzzy membership mtrix etc fuzzy membership mtrix is used i medicl digosis d decisio mkig So, if we defie orm o fuzzy membership mtrix the it will tke effective cotributio o medicl sciece REFERENCES Kdk, Bhowmik d Pl, pplictio of geerlized ituitioistic fuzzy mtrix i multi-criteri decisio mkig problem, J th Comput Sc () (0) 9-3 Kdk, Pl d Bhowmik, Distributive lttice over ituitioistic fuzzy mtrices, J Fuzzy themtics,() (03) Kdk, Bhowmik d Pl, Ituitioistic fuzzy block mtrix d its someproperties, J ls of Pure d pplied themtics, () (0) Kdk, Bhowmik d Pl, Some properties of geerlized ituitioistic fuzzy ilpotet mtrices over distributive lttice, J Fuzzy If Eg 4(4)(0) Bhowmik, Pl d Pl, Circult trigulr fuzzy umber mtrices, J Physicl Scieces, (008) Bhowmik d Pl, Ituitioistic eutrosophic set,j Iformtio d Computig Sciece, 4() (009) Bhowmik d Pl, Ituitioistic eutrosophic set reltios d some of its properties, J Iformtio d Computig Sciece, 5(3)(00) Bhowmik d Pl, Geerlized itervl-vlued ituitioistic fuzzy sets, J Fuzzy themtics, 8() (00)

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