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1 (wwwrdoderresearchco) Volue II, Issue II, 2016 PRODUC OPERAION ON FUZZY RANSIION MARICES V Chiadurai*, S Barkavi**, S Vayabalaji*** & J Parthiba**** * Departet of Matheatics, Aaalai Uiversity, Aaalai Nagar, Chidabara, ailadu ** Departet of Matheatics, Aaalai Uiversity, Aaalai Nagar, Chidabara, ailadu *** Departet of Matheatics, Uiversity College of Egieerig (A Costituet College of Aa Uiversity - Cheai), Paruti, ailadu **** Departet of Matheatics, Goveret College of Egieerig, Sriraga, iruchirappalli, ailadu Abstract: Fuzzy Matrices over fuzzy algebra plays a sigificat role i atrix theory over fuzzy settigs Fuzzy trasitio atrix is aother iterestig structure i fuzzy atrix theory Our ai i this paper is to defie product operatio o fuzzy trasitio atrix ad to provide soe iterestig result o it 1 Itroductio: Fuzzy set theory was proposed by Lotifi A Zadeh [11 ad it has extesive applicatios i various fields Fuzzy atrix was itroduced by hoaso [7 ad the cocept of ucertaity was discussed by usig fuzzy atrices Differet cocepts ad ideas of fuzzy atrices have bee give earlier aily by Ki, Meeakshi ad hoso [1, 2, 7 Fuzzy atrix plays a vital role i fuzzy odelig, fuzzy diagosis ad fuzzy cotrols It also has applicatios i fields like psychology, edicie, ecooics ad sociology It is well kow that the atrix forulatio gives a extraordiary advatage to hadle ay data or atheatical probles uder cosideratio I other words, whe soe atheatical probles are ot solved by the classical atrices, the the cocept of fuzzy atrix is used It is sure that fuzzy atrices have advatageous applicatios over fuzzy algebra ad give decisio akig results i a effective aer A Stochastic process is a atheatical odel that evolves over tie i a probabilistic aer A special kid of stochastic process is called Markov chai rasitio probability atrix plays a iportat role i Markov process Its applicatios i edicie are quite coo ad have becoe a stadard tool i decisio akig [3, 4, 5, 6 I the early ivestigatio, Vayabalaji etal studied the algebraic tool structure i fuzzy atrix ad furished ew defiitio aely fuzzy trasitio atrix [8 ad provided soe result o it I their aother fidigs, they itroduced orthogoal cubic trasitio atrices usig ier product [10 Further, they also discussed various operatios of iterval valued ituitioistic fuzzy trasitio atrices [9 I the preset study, by takig ad worked out the advatageous features of the theories of fuzzy atrices ad fuzzy trasitio atrices the ew defiitio of Product operatio o fuzzy trasitio atrix has bee itroduced Further, soe properties of trace ad traspose are also discussed I additio, soe results are explaied with a exaple i a iterestig aer Key Words: Fuzzy algebra, fuzzy atrices, fuzzy trasitio atrices & product operatio o fuzzy trasitio atrix 1 Prelaries: I this sectio we recall soe basic defiitios ad results which will be eeded 281

2 (wwwrdoderresearchco) Volue II, Issue II, 2016 i the sequel Defiitio 11: [2 By a fuzzy atrix we ea a atrix over the fuzzy algebra Defiitio 12: [2 Let V deote set of all fuzzy atrices over the fuzzy algebra he operatios (+, ) are defied o V as follows: (1) For ay two eleets A [ a ad B [ b V Defie A B ( sup a, b ) ( a, b ) where each a, b (2) For ay eleet A ( a ) V ad a scalar k F Defie ka ( if k, a ) ( k, a ) F he syste V together with these operatios of copoet wise fuzzy additio ad fuzzy ultiplicatio is called fuzzy vector space over F Defiitio 13: [2 Defie two operatios + ad o the V as follows (1) For ay two eleets A [ a ad B [ b V Defie (2) A B ( sup a, b ) ( a, b ) where each a, b AB ( Sup if a, b ) ( a, b ) F Defiitio 14: [2 For A the traspose is obtaied by iterchagig its rows ad colus ad is deoted by A Defiitio 15: [8 Let V2 2 the set of all 2 2 trasitio atrix over the fuzzy algebra he operatio (+, ) are defied o V2 2as follows For ay two eleets A [ a ad B [ b V2 2 a b,ifa b 1 () i AB a b 1,ifa b 1 [1,0 12,ifa b 1 i row 1 ( ) A B [0,1 12,ifa b 1 i row 2 ( i) For ay scalar k (0,1), ka fratioal part of (10 ka ) he syste V 2 2 together with these operatios of copoet wise trasitio additio ad ultiplicatio is called as fuzzy trasitio atrix It is deoted by ( V2 2, ) Exaple 16: [8 Let A ad B be the fuzzy trasitio atrices over he AB is a fuzzy trasitio atrix 2 Product Fuzzy rasitio Matrices: We ow eter ito our ew otio of product fuzzy trasitio atrix as follows a11 a12 a1 b11 b12 b1 a21 a22 a 2 b21 b22 b 2 Defiitio 21: Let [ a ad[ b ai1 ai 2 ai bi1 bi 2 bi a 1 a2 a b 1 b 2 b 282

3 (wwwrdoderresearchco) Volue II, Issue II, 2016 be two x fuzzy trasitio atrices he the product of the is defied by, a1 k (, ) bk 1 a1 k (, ) bk 2 a1 k (, ) bk k 1 k 1 k 1 a2 k (, ) bk 1 a2 k (, ) bk 2 a2 k (, ) bk ( i)[ a [ b k 1 k 1 k 1 ak (, ) bk 1 ak (, ) bk 2 ak (, ) bk k 1 k 1 k 1 ( ) If i the i th row all the correspodig suatio values are equal to 1 the, aik (, ) bki 1, if i k k 1 [ a [ b aik (, ) bki 0, if i k k 1 wherei 1,2,, ak bk if k is odd where (, ) ak bk if k is eve Exaple 22: Let A [ a, B [ b he the product of [ a [ b a11 a12 a1 b11 b12 b1 a21 a22 a 2 b21 b22 b 2 Defiitio 23: Let [ a ad [ b ai1 ai 2 ai bi1 bi 2 bi a 1 a2 a b 1 b 2 b be two fuzzy trasitio atrices he the product of the is defied by, 283

4 (wwwrdoderresearchco) Volue II, Issue II, 2016 a1 k (, ) bk 1 a1 k (, ) bk 2 a1 k (, ) bk k 1 k 1 k 1 a2 k (, ) bk 1 a2 k (, ) bk 2 a2 k (, ) bk ( i)[ a [ b k 1 k 1 k 1 ak (, ) bk 1 ak (, ) bk 2 ak (, ) bk k 1 k 1 k 1 ( ) If i the i th row all the correspodig suatio values are equal to 1 the, aik (, ) bki 1, if i k k 1 [ a [ b aik (, ) bki 0, if i k ak bk if k is odd where k 1 (, ) ak bk if k is eve wherei 1,2,, Exaple 24: Let [ a,[ b he the product of [ a [ b 0 01 Reark 25: he product of two fuzzy trasitio atrices is ot a fuzzy trasitio atrix Exaple 26: Let [ a ,[ b he their product [ a [ b is ot a fuzzy trasitio atrix race ad raspose of Fuzzy rasitio Matrices: Defiitio 31: Let A [ a be a fuzzy trasitio atrix of order he trace of fuzzy trasitio fuzzy atrix is tr a a11 a22 a33 a i Exaple 32: Let A[ a be a fuzzy trasitio atrix of order the trace of this atrix is tr Defiitio 33: Let A [ a be a fuzzy trasitio atrix of order he the traspose of [ a is defied by [ a [ aji 284

5 (wwwrdoderresearchco) Volue II, Issue II, Exaple 34: Let [ a , the[ a Defiitio 35: Let A [ a be a fuzzy trasitio atrix of order is said to be a syetric if a aji for all i ad j Propositio 36: Let A be the fuzzy trasitio square atrix of order ad k (0,1) be a scalar he ( i) tr( k) k( tr ) ( ) tr tr (i) Let A [ a be a fuzzy trasitio atrix of order ( i e) A [ a ka [ ka tr( k ) ka i1 k a i1 k( tr ) () Let A [ a tr a a a a a i1 A [ a B [ b ji tr tr b b b b b i a a a a a tr i1 Hece, tr tr Propositio 37: Let A ad B be two fuzzy trasitio atrices of order, the tr tr Let A [ a, B [ b be two fuzzy trasitio atrices of order he, A B [ c i1 tr( ) c c c c c B A [ d

6 (wwwrdoderresearchco) Volue II, Issue II, 2016 tr( ) d d d d d i Sice, c c c c d d d d i1 i c d Hece, tr( ) tr( ) Propositio 38: Let A [ a kl, (0,1) he the followig results hold () i ka k A ( ) k l A ka la be a fuzzy trasitio atrix of order ad let heore 39: he traspositio operatio is reflexive, that is A Let A [ a ad A [ b he [ b [ a for all pairs (i, j) he traspose of A is [ b ad thus [ a Hece A A Propositio 310: he su of the fuzzy trasitio atrix of order ad its traspose is a syetric atrix ( i e) A A is a syetric atrix Let A [ a be a fuzzy trasitio atrix of order he A A [ b B Sice [ b [ b [ b ( i e) B B ji Hece B is a syetric atrix heore 311: he traspose of the su of the two fuzzy trasitio atrices of order is equal to the su of their trasposes, ( i e)( A B) A B B A Let A [ a, B [ b be two fuzzy trasitio atrices of order he A B C, C [ c where [ c [ a [ b for all pairs (i, j) Let A [ a, B [ b he [ ji A B c ji ji aji b ji A B Hece ( A B) A B Reark 312: I geeral A1 A2 A A1 A2 A heore 313: he traspose of the product of two fuzzy trasitio atrices is equal to the product of their traspose i reverse order ( i e)( A B) B A Let A [ a, B [ b be two fuzzy trasitio atrices of order Let AB [ c A 286

7 (wwwrdoderresearchco) Volue II, Issue II, 2016 he [ c aik (, ) bkj ad is the eleets of the j th row ad i th colu of [ c k1 ( i e)( AB ) th th he eleets b1j, b2j,, bj of the j colu of B are the eleets of j row of he eleets ai 1, ai 2,, ai of the i th row of A are the eleets of i th colu of he eleets of th j row ad hat is aik (, ) bkj k1 th i colu of B A is bkj (, ) a k1 Hece, ( A B) B A heore 314: he su of ay fuzzy trasitio atrix ad its traspose is a syetric atrix; that is ( A A ) ( A A ) ( A A ) A ( A ) ; by heore 311 A A; by heore 39 A A sice the additio of atrices is coutative Hece A A is a syetric atrix heore 315: Let A [ a be a fuzzy trasitio atrix of order 2, the ( A A) A A, is a syetric atrix ( A A) A ( A ) sice( AB) B A A A sice( A ) A Hece, AA is a syetric atrix Reark 316: If A is a fuzzy trasitio atrix of order 2, the AA is a idetity atrix of order ; ad AA is a syetric atrix of order 2 It is show by the followig theores heore 317: he product of AA of two fuzzy trasitio atrices always a idetity atrix ( i e)[ a 2[ a ji 2 I idetity atrix a11 a12 a21 a 22 Let A[ a, the the traspose of [ a is a1 a2 2 a11 a12 a 1 A [ a a21 a22 a 2 2 he the product of AA = [ a [ a is, B A 287

8 (wwwrdoderresearchco) Volue II, Issue II, 2016 a11 a12 a21 a 22 a11 a12 a 1 [ a [ a, a21 a22 a 2 2 a1 a2 2 a1 k (, ) bk 1 a1 k (, ) bk 2 a1 k (, ) b k k 1,2 k 1,2 k 1,2 a2 k (, ) bk 1 a2 k (, ) bk 2 a2 k (, ) bk k 1,2 k 1,2 k 1,2 = ak (, ) bk 1 ak (, ) bk 2 ak (, ) bk k 1,2 k 1,2 k 1, = I a Idetity atrix [ sice by the defiit io 23 () heore 318: he product of AA of two fuzzy trasitio atrix is always a syetric atrix a11 a12 a21 a 22 Let A[ a, the the traspose of [ a is a1 a2 2 a11 a12 a 1 A [ a a21 a22 a 2 2 he the product of A A [ a [ a is, a11 a12 a 11 a12 a 1 a21 a 22 [ a [ a = a21 a22 a 2 2 a a ak1 (, ) ak1 ak1 (, ) a2k k1 k1 = a2 k (, ) ak1 a2 k (, ) a2k k1 k1 = [ b 2 2 Bisa2 2 syetric atrix [Sice B B heore 319: If A is a fuzzy trasitio atrix of order, the pre ad post product of A with A is a fuzzy trasitio atrix of order ( i e) AA A is a fuzzy 288

9 (wwwrdoderresearchco) Volue II, Issue II, 2016 trasitio atrix of order Let A [ a, the the traspose of [ a is [ aji A AA [ a [ a AA I, by the heore 317 AA A ( AA ) A I A I [ a [ b B Hece B is a fuzzy trasitive atrix of order Refereces: 1 Ki JB, Idepotet ad Iverses i Fuzzy Matrices, Malaysia Math, 6(2), 57-61, Meeakshi AR, Fuzzy Matrix heory ad Applicatios, MJP Publishers, Peebles PZ, Jr, Probability, Rado Variables ad Rado Sigal Priciples, Megraw Hill Copay, New York, Ross S, Itroductio to Probability Models 8 th ed, Acadeic Press, Sa Diego, Califoria, Rudi W, Fudaetal Aalysis, McGraw Hill Sciece uckwell HD, Eleetary Applicatios of Probability heory, 2 d Ed, Chapa ad Hall, Lodo, hoaso MG, Covergece power of fuzzy atrix, J of Math Aal Appl 57, , Vayabalaji S, Shyasudar G, Parthiba J & Narayaa, Fuzzy operatios o trasitio atrices, Global Joural of Pure ad Applied Matheatics, Volue 12, No: 3, , Vayabalaji S & Shyasudar G, Iterval-valued ituitioistic fuzzy trasitio atrices, Iteratioal Joural of Moder Sciece ad echology, Vol 1, Issue 2, 47-51, May Vayabalaji S & Shyasudar G, Cubic rasitio Matrices, Asia Joural of Research i Social Scieces ad Huaities, Vol 6, No 7, , July Zadeh LA, Fuzzy sets, Iforatio ad Cotrol 8, ,