ON THE HADAMARD PRODUCT OF BALANCING Q n B AND BALANCING Q n

Transcription

1 TWMS J App Eg Math V5, N, 015, pp ON THE HADAMARD PRODUCT OF ALANCING Q AND ALANCING Q MATRIX MATRIX PRASANTA KUMAR RAY 1, SUJATA SWAIN, Abstract I this paper, the matrix Q Q which is the Hadamard product of both balacig Q matrix ad balacig Q matrix is itroduced Some properties of the Hadamard product of these matrices are ivestigated A differet codig ad decodig method based o the applicatio of the Hadamard product of balacig Q matrix ad balacig Q matrix is also cosidered Keywords: alacig umbers, alacers, alacig Q-matrix, Cryptography AMS Subject Classificatio: 11 39, 11 83, 11T 71 1 Itroductio The balacig umbers are the terms of the sequece {0, 1, 6, 35, 0, } ad their recurrece relatio is give by , 1, 1 with iitials 0 0 ad 1 1 [1] May importat ad useful results of these umbers ad their related sequeces are available i the literature Iterested reader ca go through [, ] There is aother way to geerate balacig umbers usig powers of a matrix called as balacig Q-matrix itroduced by Ray i [13] The balacig matrix is a secod order matrix whose etries are the first three balacig umbers 0, 1 ad 6, ad is i the form Q I [13], he has also show that the th power of the balacig Q-matrix is i the form Q +1, 1 with the determiat value 1, ie by Cassii formula for balacig umbers, detq The recurrece relatio 1 ca be used to exted the balacig umbers backward to get 1 Veer Suredra Sai Uiversity of Techology, Odisha, urla, Idia rayprasata008@gmailcom; DAV Uit-8 hubaeswar, Idia sujatasw@gmailcom; Mauscript received: March 11, 01 TWMS Joural of Applied ad Egieerig Mathematics, Vol5, No; c Işık Uiversity, Departmet of Mathematics, 015; all rights reserved 01

2 0 TWMS J APP ENG MATH V5, N, 015 We ow preset some basic results relatig to the th power of the balacig Q-matrix, Q Lemma 11 The balacig matrix Q is also satisfy the recurrece relatio 1 of the balacig umbers, that is Q 6Q 1 Q Proof The proof is easy y 1, we obtai Q which completes the proof 6Q 1 Q, Lemma 1 The followig property for Q is valid: Q Qm Qm Q Q+m Proof Sice +1 m m 1 m+1 m 1 m+ [11], we have Q Q m +1 m+1 m 1 m m 1 +1 m+1 m +1 m + m 1 m+1 m 1 m + 1 m 1 +m+1 +m +m +m 1 Q +m Other part ca be show similarly I this study, we will cosider the Hadamard product of balacig Q matrix ad balacig Q matrix deoted by Q Q, where Q is the iverse of the matrix We will also ivestigate some importat properties of this product Q Some idetities of Q Q y virtue of, the Hadamard product Q Q Q Q Q adjq where adjq is the adjoit of the matrix Q The followig defiitio is give i [3, 1] matrix ca be writte as +1 1, +1 1 Defiitio 1 Let A a ij be matrix over ay commutative rig The permaet of A deoted by pera is defied by pera a 1σ1 a σ a σ, σ where the summatio exteds over all oe-to-oe fuctios from {1,,, } to {1,,, } The followig are some importat results o the Hadamard product Q Q

3 PRASANTA KUMAR RAY AND SUJATA SWAIN: ON THE HADAMARD PRODUCT OF 03 Theorem 1 For all itegers, det Q Q 1 Proof Usig Defiitio 1 ad the idetity 3, we get det Q Q which eds the proof perq 1, The followig corollary is a immediate cosequece of Theorem 1 Corollary 1 The trace of the matrix Q Q is, trace Q Q 1 Theorem If λ 1 ad λ are the eigevalues of the matrix Q Q, the λ 1 1, λ perq Proof Let I is the idetity matrix of order y 3, the characteristic equatio of the matrix Q Q is give by 0 det Q Q λi λ λ λ λ + perq λ 1 It follows that λ 1 1 ad λ perq Theorem 3 The liearly idepedet eigevectors correspodig to the eigevalues 1 1 λ 1 1 ad λ perq of the matrix Q Q are X λ 1 ad X 1 λ 1 Proof If λ is a eigevalue of the matrix Q Q, the the correspodig eigevectors x1 X λ are the solutio of the equatio x Q Q λi X λ 0 5 For λ 1 1, 5 reduces to x x 0 Usig 3 agai, we obtai x1 x 0 0, which is a system of homogeous equatios Therefore by elemetary row operatio, we get 1 1 x x 0 Sice the rak of the coefficiet matrix of this system is 1, there exists ifiitely may solutios depedig o oe parameter Therefore, the solutios of the system are x 1

4 0 TWMS J APP ENG MATH V5, N, 015 k, x k, where k is arbitrary Hece, the liearly idepedet eigevector correspodig to the eigevalue λ 1 1 is equal to [ 1, 1] T Similarly, For λ perq ad by 3 agai, 5 reduces to x1 0 x 0 Oe ca proceed similarly to get x 1 x k, where k is arbitrary Thus, the liearly idepedet eigevector correspodig to the eigevalue λ perq is equal to [1, 1]T Which completes the proof Remark 1 Sice the matrix Q Q is symmetric, it ca be diagoalize Therefore by virtue of Theorem ad Theorem 3, we ca write the matrix P i the form 1 1 P ad otice that, P Q Q P diag1, perq It is well kow that, if M deote the class of complex matrices, the the maximum colum sum matrix orm o M is defied by A 1 max a ij 1 j ad the maximum row sum matrix orm o M is defied by A max a ij i1 1 i j1 Also, the l 1 orm ad the Euclidea orm or l orm o M are respectively give by A 1 a ij ad A 1,j1 1,j1 a ij The followig idetities are easily deduced from the defiitio of orms Theorem For all itegers, we have Theorem 5 The matrix Q Q a Q Q 1 Q Q 1, b Q Q 1, c Q Q + is ivertible, ad Q Q Proof y virtue of Theorem, det Q Q perq 1 0 Therefore it 1 1 is ivertible, ad its iverse ca be easily deduced as Q Q This eds the proof

5 PRASANTA KUMAR RAY AND SUJATA SWAIN: ON THE HADAMARD PRODUCT OF 05 3 alacig codig/decodig method I this sectio, we cosider a simple codig/decodig method based o applicatio of the Hadamard product Q Q Let the iitial massage M is represeted by a matrix of the form m1 m M m 3 m ased o matrix multiplicatio, we ow cosider the followig ecryptio/decryptio algorithms Ecryptio: Decryptio: M Q Q E Ex Q 1 Q M We assume that the etries of M are all positive itegers, ie m 1 > 0, m > 0, m 3 > 0, m > 0 To describe the method, for example we select the matrix Q 3 Q 3 as the codig matrix The Q 3 Q ad Q 3 Q Thus the balacig codig of the massage M cosists i its multiplicatio by the direct codig matrix 6, that is M Q 3 Q 3 m1 m 1 15 m 3 m m1 15m 15m 1 1m 1m 3 15m 15m 3 1m e1 e E, e 3 e where e 1 1m 1 15m, e 15m 1 1m, e 3 1m 3 15m, e 15m 3 1m Thus, the set code massage E {e 1, e, e 3, e } is ow decoded by multiplyig it with the iverse matrix 7 i the followig way: e1 e e 3 e e e 1 9 e e e e 15 9 e e e 1 e e 3 e y simple algebraic maipulatio with the help of the idetities e 1, e, e 3 ad e, oe ca easily obtai e 1 e m1 m e 3 e M m 3 m

6 06 TWMS J APP ENG MATH V5, N, 015 We otice that, the determiat of the code matrix E which is obtaied from the multiplicatio of iitial matrix M with the codig matrix Q Q is give by det E det M Q Q 1, for all itegers Refereces [1] ehera, A ad Pada, GK, 1999, O the square roots of triagular umbers, 37, Fiboacci Quart, pp [] èrczes, A, Liptai, K ad Pik, I, 010, O geeralized balacig umbers, Fiboacci Quart, 8, pp [3] Hor, RA ad Johso, CA, 1985, Matrix Aalysis, Cambridge Uiversity Press, New York [] Keski R ad Karaatly O, 01, Some ew properties of balacig umbers ad square triagular umbers, JIteger Seq, 151, pp 11 [5] Liptai, K, Fiboacci balacig umbers, 00, Fiboacci Quart,, pp [6] Liptai, K, Lucas balacig umbers, 006, Acta Math Uiv Ostrav, 11, pp 3-7 [7] Liptai, K, Luca, F, Piter A ad Szalay L, 009, Geeralized balacig umbers, Idag MathN S, 0, pp [8] Olajos, P, 010, Properties of balacig, cobalacig ad geeralized balacig umbers, A Math Iform, 37, pp [9] Pada, GK ad Ray PK, 011, Some liks of balacig ad cobalacig umbers with Pell ad associated Pell umbers, ull Ist Math Acad Si N S, 61, pp 1-7 [10] Pada, GK ad Ray PK, 005, Cobalacig umbers ad cobalacers, It J of Math Math Sci, 8, pp [11] Pada, GK 009, Some fasciatig properties of balacig umbers, Proceedig of the Eleveth Iteratioal Coferece o Fiboacci Numbers ad Their Applicatios, Cogr Numer, 19, pp [1] Patel, K ad Ray, PK 015, The Period, rak ad order of the sequece of balacig umbers modulo m, accepted i Mathematical Reports [13] Ray PK, 01, Applicatio of Chybeshev polyomials i factorizatio of balacig ad Lucasbalacig umbers, ol, Soc Paraa Mat, 30, pp 9-56 [1] Ray PK, 01, Certai matrices associated with balacig ad Lucas-balacig umbers, Matematika, 8 1, pp 15- [15] Ray PK, 013, Factorizatio of egatively subscripted balacig ad Lucas-balacig umbers, ol, Soc Paraa Mat, 31, pp [16] Ray PK, 01, Curious cogrueces for balacig umbers, It J of Cotemp Math Sci, 7 18, pp [17] Ray PK, 013, New idetities for the commo factors for balacig ad Lucas-balacig umbers, It J Pure Appl Math, 85, pp 87-9 [18] Ray PK, 01, Some cogrueces for balacig ad Lucas-balacig umbers ad their applicatios, Itegers, 1, #A8 [19] Ray PK, 01, O the properties of Lucas-balacig umbers by matrix method, Sigmae, Alfeas, 31, pp 1-6 [0] Ray PK, Parida K, 01, Geeralizatio of Cassii formula for balacig ad Lucas-balacig umbers, Matematychi Studii, 1, pp 9-1 [1] Ray PK, Dila GK, Patel K, 01, Applicatio of some recurrece relatios to cryptography usig fiite state machie Iteratioal Joural of Computer Sciece ad Electroics Egieerig IJCSEE,, pp 0-3 [] Ray PK, 01, Idetities ivolvig the terms of a balacig-like sequece via matrices, Caspia Joural of Applied Mathematics, Ecology ad Ecoomics, 1, pp [3] Ray PK, 015, alacig ad Lucas balacig sums by matrix methods, Mathematical Reports, 1767,, pp 5-33 [] Ray, PK ad Patel, K 015, Uiform distributio of the sequece of balacig umbers modulo m, accepted i Uiform Distributio Theory

7 PRASANTA KUMAR RAY AND SUJATA SWAIN: ON THE HADAMARD PRODUCT OF 07 Sujata Swai received her MCA degree from iju Pattaik Uiversity ad Techology, Roukela, Idia ad M Tech From erhampur Uiversity, Idia She is curretly workig as a computer teacher i the Departmet of Computer Sciece at DAV Public School, Uit-8, hubaeswar, Idia Her research iterests are i the areas of Number Theory ad Cryptography, Parallel Algorithm, Artificial Itelligece ad Neural Networkig She has more tha two papers i her credit Prasata Kumar Ray received his PhD from NIT Rourkela, Idia ad is curretly workig as a Associate Professor i the Departmet of Mathematics at Veer Suredra Sai Uiversity of Techology, Odisha, urla, Idia His research iterests icludes Number Theory ad Cryptography He has published more tha 5 coferece/joural articles