Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 35, Fermat-GCD Matrices

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1 Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 35, Fermat-GCD Matrices Şerife Büyükköse The University of AhiEvran, Faculty of Science and Arts Department of Mathematics, 4000 Kirşehir, Turkey Dursun Taşci Gazi University, Faculty of Sciences and Arts Departments of Mathematics, Teknikokullar Ankara, Turkey Abstract. In this paper we defined an n n matrix [F ]f ij where f ij 2 2 x i, +, Fermat-GCD Matrix on FC set S {x,x 2,...,x n }. We calculated the determinant and the inverse of the Fermat-GCD Matrix on FC set S by arithmetic functions. Mathematics Subject Classification: Primary 05C38, 5A5; Secondary 05A5, 5A8 Keywords: Gcd matrices, Fermat numbers. Introduction Let S {x,x 2,...,x n } be ordered set of distinct positive integers.the n n matrix [S] s ij, where s ij x i,, the greatest common divisor of x i and is called the greatest common divisor GCD matrix on S []. In 876, H. J. S. Smith [2] showed that the determinant of the GCD matrix defined on S {, 2,...n} Smith s determinant is equal to φφ2...φn, where φ is Euler s totient function. The set S is factor-closed FC if it contains every divisor of x for any xɛs. In this paper we define an n n matrix [F ]f ij, where f ij 2 2i,j +, call it to be the Fermat-GCD matrix on S {, 2,..., n}. In the second section we calculate the determinant, the trace and the inverse of the Fermat-GCD matrix on set S by the arithmetical function g and, Möbiüs function.

2 746 Ş. Büyükköse and D. Taşci 2. The Structure of the Fermat-GCD Matrix Definition. Let S {x,x 2,...,x n } be a set of distinct positive integers and the n n matrix [F ]f ij, wheref ij 2 2i,j +, call it to be Fermat-GCD Matrix on S. Theorem. Let S {, 2,..., n} and define the m m matrix Ψdiag gd,gd 2,...,gd m, where gn n 2 2d + d d n and n m matrix H h ij by { if d i e ij Then [F ]HΨH T. Proof. The ij-entry in HΨH T is n HΨH T h ij ik Ψ k h kj k gd k d k x i d k gd k d k x i, 2 2i,j +f ij Theorem 2. Let S {d,d 2,...,d m } be the minimal ordered FC set containing S {x,x 2,...,x n }.Then n det [F ] g i...g n i where H k,...,k n is the submatrix of H consisting of k th,..., k n th columns of H. T Proof. We says that; [M] HΨ 2 HΨ 2 and Cauchy-Binet formula we obtain; [ ] T det [F ] det HΨ 2 HΨ 2 [ det HΨ 2 k,...,kn ] 2

3 and and then; det det [F ] [ HΨ 2 Fermat-GCD matrices 747 k,...,kn 6k <k 2 <...<k n6m ] det H k,...,k n g x...g x n 2 det Hk,...,k n g x k...g x kn. Example. Consider the 3 3 Fermat-GCD matrix on FC set S {2, 3, 6} [F ] and S {, 2, 3, 6}, an3 4 matrix H h ij is [H] By using the theorem we obtain, det [F ]. Theorem 3. Let [F ] f ij is the n n -GCD matrix on FC set S {x,x 2,...,x n }.Then its inverse is the n n matrix [B] b ij such that; b ij gx k. x i x i x k x k Proof. The n n matrix [Y ]y ij defined by { x i y ij x i Calculating the ij-entry of product HY gives, HY ij n h ik y kj x k x k k Thus H Y. Since [F ]HΨH T and H Y { i j x k x k x i then, [F ] HΨH T Y T Ψ 2 Y bij

4 748 Ş. Büyükköse and D. Taşci where b ij x i x k x k gx k x i. Thus, the proof is complete. Example 2. Let [F ]f ij is a Fermat-GCD matrix on FC set S {, 2, 3, 6}. Then, [F ] [B] b ij where b 2 g + 22 g g g6 9 b 2 2 g2 b 3 3 g3 b 4 6 g g g6 b 22 2 g g6 28 b g6 b 24 3 g6 b 33 2 g g6 0 b 34 2 g6 b 44 2 g Therefore, since [F ] B is the symmetric we have

5 2.. [F ] B Fermat-GCD matrices The Structure of the Reciprocal Fermat-GCD Matrix. Definition 2. Let S {x,x 2,...,x n } be a set of distinct positive integers and the n n matrix [R] r ij, where r ij, call it to be Reciprocal 2 2x i, + Fermat-GCD Matrix on S. Theorem 4. Let S {x,x 2,...,x n } be an ordered set of distinct positive integers and S {d,d 2,...,d m } the minimal FC ordered set containing S. Define the m m matrix Ψdiag βd,βd 2,...,βd m, where βn n 2 2d + d d n and n m matrix Q q ij by { if dj x q ij i Then [R] QΨQ T. Proof. The ij-entry inqψq T is n QΨQ T q ij ik Ψ k q kj k βd k d k x i d k βd k d k x i, 2 2 x i, + r ij Theorem 5. Let S {d,d 2,...,d m } be the minimal ordered FC set containing S {x,x 2,...,x n }.Then 2 det [R] det Qk,...,k n β x k...β x kn 6k <k 2 <...<k n6m

6 750 Ş. Büyükköse and D. Taşci where Q k,...,k n is the submatrix of Qconsisting of k th,..., k n th columns of Q. T QΨ 2 QΨ 2 and Cauchy-Binet formula we Proof. We says that; [R] obtain; and and then; det det [R] [ QΨ 2 det [R] det k,...,kn 6k <k 2 <...<k n6m [ ] T QΨ 2 QΨ 2 [ det QΨ 2 k,...,kn ] 2 ] det Q k,...,k n β x...β x n 2 det Qk,...,k n β...β x kn. Example 3. Consider the 3 3 Reciprocal Fermat-GCD matrix on FC set S {2, 3, 6} [R] and S {, 2, 3, 6}, an3 4 matrix Q q ij is [Q] By using the theorem we obtain, det [R] Theorem 6. Let [R] r ij is the n n Reciprocal Fermat-GCD matrix on FC set S {x,x 2,...,x n }.Then its inverse is the n n matrix [B] b ij such that; b ij x i x k x k βx k x i Proof. The n n matrix [Y ]y ij defined by { x i y ij x i.

7 Calculating the ij-entry of product QY gives, Fermat-GCD matrices 75 QY ij n h ik y kj k x k x k { i j x k x k x i Thus Q Y. Since [R] QΨQ T and Q Y then, [R] QΨQ T Y T Ψ 2 Y bij where b ij x i x k x k βx k x i. Thus, the proof is complete. Example 4. Let [R] r ij is a Reciprocal Fermat-GCD matrix on FC set S {, 2, 3, 6}. Then, [R] [B] b ij

8 752 Ş. Büyükköse and D. Taşci where b 2 β + 22 β β β6 b 2 2 β2 b 3 3 β3 b 4 6 β β β6 b 22 2 β β6 b β6 b 24 3 β6 b 33 2 β β6 b 34 2 β6 b 44 2 β6 Therefore, since [R] B is the symmetric we have [R] B References [] P. Haukkanen and J. Sillanpaa, Some Analogues os Smith s Determinant, Linear and Multilinear Algebra, 4996, [2] P.J. McCarthy, Introduction to Arithmetic Functions, New York, Springer Verlag 986 [3] R.T.Hansen and L.G.Swanson, Unitary Divisors, Math, Mag.,52 979, [4] S. Buyukkose and D. Tasci, On The Mersenne GCUD Matrices, Intern. Math. Journal, Vol.3 No.,2003,0-05 [5] S. Buyukkose and D. Tasci, On The Reciprocal Mersenne GCUD Matrices, Intern. Math. Journal, Vol.3 No.6,2003, [6] S. Beslin and S. Ligh, Greatest Common Divisor Matrices, Linear Algebra and Its Applications, 8: [7] T.M.Apostol, An Introductıon to Analytıc Number Theory. st Ed. New York Springer Verlag, 976

9 Fermat-GCD matrices 753 [8] K. Bourque and S.Ligh, On GCD and LCM Matrices,Linear Algebra and Its Appl.74:992, [9] S. Beslin, Reciprocal GCD Matrices And LCM Matrices, Fibonacci Quarterly, 29: [0] H.J.S. Smith, On The Value of a Certain Arithmetical Determinant, Proc. London Math.Soc , [] M. Rosen, Number Theory in Function Field, GTM 20,New York Springer Verlag,2002 Received: September 2, 2008