On the Hadamard Product of Fibonacci Q n matrix and Fibonacci Q n matrix

Transcription

1 Int J Contemp Math Sciences, Vol, 006, no 6, On the Hadamard Product of Fibonacci Q n matrix and Fibonacci Q n matrix Ayşe NALLI Department of Mathematics, Selcuk University 4070, Campus-Konya, Turkey aysenalli@yahoocom Abstract In this paper we defined, the matrix Q n Q n, Hadamard product of Fibonacci Q n matrix and Fibonacci Q n matrix We investigated some properties of Hadamard product of Fibonacci Q n matrix and Fibonacci Q n matrix Mathematics Subject Classification: Primary, B5, B37, B39, Secondary, C0 Keywords: Fibonacci Q matrix, Fibonacci Q n matrix, Fibonacci Q n matrix Introduction In the last decades the theory of Fibonacci numbers,3 was complemented by the theory of the so-called F ibonacci Q matrix,3 The latter is a square matrix of the following form, Q 0 In 3 the following property of the n th power of the Q matrix was proved ) Q n ) ) Fn+ F n F n F n ) detq n ) F n+ F n Fn ) n where n 0, ±, ±,, F n, F n, F n+ are Fibonacci numbers given with the following recurrence relation : 3) F n+ F n with the initial terms F F + F n

2 754 Ayşe NALLI Rule 3) can be used to extend the sequence backwards, thus F n ) n+ F n In 7, represent matrix ) is showed in the following form Q n or ) Fn + F n F n + F n F n + F n F n + F n 3 ) ) Fn F n Fn F + n F n F n F n F n 3 Q n Q n + Q n It is proved in 8 the following property of the Q matrix, Q n Q m Q m Q n Q n+m In this paper we defined the matrix Q n Q n, Hadamard product of Fibonacci Q n matrix and Fibonacci Q n matrix, where Q n is Fibonacci matrix in ) and Q n is inverse matrix of Fibonacci Q n matrix We investigated some properties of Hadamard product of Fibonacci Q n matrix and Fibonacci Q n matrix Some Properties of the Q n Q n Matrix Let Q n be Fibonacci matrix Q n is inverse of Q n Fibonacci matrix Then Hadamard product of Q n Fibonacci matrix and Q n Fibonacci matrix, Q n Q n is defined by Q n Q n { Q n adjq n neven Q n adjq n ) n odd Definition 5 Let A a ij ) be n n matrix over any commutative ring The permanent of A, written pera),is defined by pera) σ a σ a σ a nσn where the summation extends over all one-to one functions from {,,, n} to {,,, n} { + F Theorem det Q n Q n ) k, n k Fk+, n k +

3 Proof Let n k Hadamard Product of Fibonacci matrix 755 Q n Q n Fk+ F k Fk Fk F k+ F k det Q n Q n Fk+ F ) det k Fk Fk F k+ F k ) F k+ F k Fk) Fk+ F k + Fk ) k perq n ) +Fk Let n k + Q n Q n Fk+ F k Fk+ Fk+ F k+ F k det Q n Q n Fk+ F ) det k Fk+ Fk+ F k+ F k F k+ F k Fk+) Fk+ F k + Fk+ ) k+ perq n ) Fk+ ) Corollary traceq n Q n ) { + F n ), n even F n ), n odd Theorem 3 The eigenvalues of the matrix Q n Q n are λ,λ per Q n ) neven λ,λ per Q n ) n odd Proof Let n k The characteristic polynomial of the matrix Q n Q n is Δ Q n Q nλ) detλi Qn Q n ) λ Fk+ F det k Fk Fk λ F k+ F k ) λ F k+ F k Fk) λ Fk+ F k + Fk λ F k+ F k + Fk) λ F k+ F k Fk) λ perq n )) λ ) Hence, the eigenvalues of Q n Q n are λ,λ per Q n ) Let n k + The characteristic polynomial of the matrix Q n Q n is

4 756 Ayşe NALLI λ + Δ Q n Q nλ) detλi Qn Q n Fk+ F ) det k Fk+ Fk+ λ + F k+ F k ) λ + F k+ F k Fk+) λ + Fk+ F k + Fk+ λ ) λ + perq n )) Hence, the eigenvalues of Q n Q n are λ,λ per Q n ) Theorem 4 The linearly independent eigenvectors corresponding to the eigenvalues of the matrix Q n Q n are y, y Proof If λ i an eigenvalue of the matrix Q n Q n, the corresponding eigenvectors Y i are the solutions of ) λi I Q n Q n) Y i 0 Let n k We first calculate the eigenvector corresponding to λ Then, λ I Q n Q n) From ), Fk+ F k Fk F Fk k Fk F k+ F k Fk Fk F k F k F k F k y 0 y 0 By using elementary row operations, the coefficients matrix of this homogeneous system becomes y 0 y Since the rank of the coefficients matrix of this homogeneous system is equal to, there exist infinitely many solutions dependent on one parameter The solution to this set of equations is y y k, where k is arbitrary In this case, linearly independent eigenvector corresponding to λ is equal to T Now calculate the eigenvector to λ perq n ) From ),

5 Hadamard Product of Fibonacci matrix 757 F k F k F k F k y 0 y 0 By using elementary row operations, the coefficients matrix of this homogeneous system becomes 0 0 y 0 y 0 Since the rank of the coefficients matrix of this homogeneous system is equal to, there exist infinitely many solutions dependent on one parameter The solution to this set of equations is y k, y k, where k is arbitrary In this case, linearly independent eigenvector corresponding to λ per Q n )is equal to T Similarly it is easily seen that for n k + the linearly independent eigenvectors corresponding to the eigenvalues of the matrix Q n Q n are y, y Remark Since the matrix Q n Q n is symmetric, it is diagonalizable In view of Theorems and Theorems 3 we write to obtain P ) P Q n Q n )P diag, per Q n ) neven P Q n Q n )P diag, per Q n ) n odd Definition 4 Let M n denote the class of complex n n matrices The maximum column sum matrix norm on M n is defined by A max j n n a ij i and the maximum row sum matrix norm on M n is defined by A the l norm on M n is defined by max i n n a ij j

6 758 Ayşe NALLI A n i, j a ij and the Euclidean norm or l norm on M n is defined by A n i, j a ij ) / i-) Qn Q n Q n Q n Fn +, n even Q n Q n Q n Q n Fn, n odd ii-) Qn Q n +4Fn, n even Q n Q n 4Fn, n odd iii-) Qn Q n 4Fn 4 +4F n +, n even Q n Q n 4Fn 4 4F n +, n odd Proof Theorem 4 follows easily from the definition of the norms Theorem 5 The matrix Q n Q n is invertible and Q n Q n) Q n Q n) +Fk +Fk Fk Fk +Fk +Fk +Fk +Fk Fk+ Fk+ Fk+ Fk+ Fk+ Fk+,nk,nk + Proof Let n k F k+ F k+ adjq n Q n Fk+ F ) k Fk Fk F k+ F k By using ) we obtain F k+ F k +Fk For adjq n Q n ) +F k F k F k +F k and from Theorem, det Q n Q n ) per Q n )+Fk, Q n Q n) +F k F +Fk k +Fk Let n k + F k +F k +F k +F k

7 Hadamard Product of Fibonacci matrix 759 adjq n Q n ) Fk+ F k Fk+ Fk+ F k+ F k By using ) we obtain F k+ F k +Fk+ For F adjq n Q n ) k+ Fk+ Fk+ Fk+ and from Theorem, det Q n Q n ) per Q n ) Fk+, Q n Q n) Fk+ Fk+ Fk+ Fk+ Fk+ Fk+ F k+ F k+ 3 Fibonacci coding / decoding method 3 Example of the Fibonacci coding / decoding method Similar to 7 let us consider now the simplest Fibonacci coding / decoding method based on applicationof the Q n Q n Let us represent the initial message M in the form of the following matrix : 3) M m m m 3 m 4 Coding M Q n Q n )E Decoding E Q n Q n ) M Let us assume that all elements of the matrix 3) are positive integers, that is m > 0, m > 0, m 3 > 0, m 4 > 0 Let us assume now that we have selected the matrix Q 5 Q 5 as the coding matrix, that is 3) 33) Q 5 Q 5 Q 5 Q 5) F6 F 4 F5 F5 F 6 F 4 F 5 F5 F5 F5 F5 F5 F5 F Then the Fibonacci coding of the message 3) consists in its multiplication by the direct coding matrix 3), that is 5 4

8 760 Ayşe NALLI 34) where M Q 5 Q 5) m m 4 5 m 3 m m +5m 5m 4m 4m 3 +5m 4 5m 3 4m 4 e e E e 3 e 4 35) e 4m +5m e 5m 4m e 3 4m 3 +5m 4 e 4 5m 3 4m 4 Then the code message E e,e,e 3,e 4 is sent to a channel The decoding of the code message E given with 34) is performed in the following manner The code message E that is represented in the matrix form 34) is multiplied by the inverse matrix 33), 36) e e e 3 e e e e e e e 3 e 4 e 4 5 e + 4e 5 e 3 + 4e 4 Calculating the elements of the matrix 36) and taking into consideration 35) we get e e e 3 e 4 m m m 3 m 4 M Theorem 6 The determinant of the code matrix E that is got as result of multiplication of the initial matrix M by the coding matrix Q n Q n is det E det M Q n Q n ) { det M F k +), n k det M Fk+ ), n k + References El Naschie MS, Statistical geometry of a cantor diocretum and semiconductors, Comput Math Appl, 995, 9),03-0 Gould HW, A History of the Fibonacci Q-matrix and a higher-dimensional problem, The Fibonacci Quart 989),50-7

9 Hadamard Product of Fibonacci matrix 76 3 Hoggat VE, Fibonacci and Lucas numbers, Palo Alto, CA : Houghton-Mifflin, Horn RA, Johnson CA, Matrix Analysis, Cambridge University Press, New York, Minc H, Permanents, In Encyclopaedia of Mathematics and Its Applications,Vol6, Addison-Wesley 978) 6 Stakhov A, Massingue V, Sluchenkova A, Introduction into Fibonacci coding and cryptography, Kharkov, Osnova,999 7 Stakhov AP, Fibonacci matrices, a generalization of the Cassini formula and a new coding theory, Chaos, Solitons and Fractals,006 8 Stakhov OP, A generalization of the Fibonacci Q-matrix, Rep Nat Acad Sci,Ukraine, 9999), Taşcı D, On the Hadamard Products of itsadjoint matrix with a square matrix, Selcuk University Journal of Science, 0007),43-9 Received: May 9, 006