Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 35, 745-753 Fermat-GCD Matrices Şerife Büyükköse The University of AhiEvran, Faculty of Science and Arts Department of Mathematics, 4000 Kirşehir, Turkey serifebuyukkose@hotmail.com Dursun Taşci Gazi University, Faculty of Sciences and Arts Departments of Mathematics, Teknikokullar Ankara, Turkey dtasci@gazi.edu.tr 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.
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
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}. 7 5 7 [F ] 5 257 257 7 257 and S {, 2, 3, 6}, an3 4 matrix H h ij is [H] 0 0 0 0 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
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 g2 + 32 g3 + 62 g6 9 b 2 2 g2 b 3 3 g3 b 4 6 g6 + 63 g6 + 62 g6 b 22 2 g2 + 32 g6 28 b 23 32 g6 b 24 3 g6 b 33 2 g3 + 22 g6 0 b 34 2 g6 b 44 2 g6 28 0 Therefore, since [F ] B is the symmetric we have
2.. [F ] B Fermat-GCD matrices 749 9 28 0 28 0 28 0 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
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}. 7 5 7 [R] 5 7 257 257 257 and S {, 2, 3, 6}, an3 4 matrix Q q ij is [Q] 0 0 0 0 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.
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
752 Ş. Büyükköse and D. Taşci where b 2 β + 22 β2 + 32 β3 + 62 β6 b 2 2 β2 b 3 3 β3 b 4 6 β6 + 63 β6 + 62 β6 b 22 2 β2 + 32 β6 b 23 32 β6 b 24 3 β6 b 33 2 β3 + 22 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, 233-244 [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, 27-222 [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.6,2003,64-645 [6] S. Beslin and S. Ligh, Greatest Common Divisor Matrices, Linear Algebra and Its Applications, 8: 69-76 989 [7] T.M.Apostol, An Introductıon to Analytıc Number Theory. st Ed. New York Springer Verlag, 976
Fermat-GCD matrices 753 [8] K. Bourque and S.Ligh, On GCD and LCM Matrices,Linear Algebra and Its Appl.74:992, 65-74 [9] S. Beslin, Reciprocal GCD Matrices And LCM Matrices, Fibonacci Quarterly, 29:27-27499 [0] H.J.S. Smith, On The Value of a Certain Arithmetical Determinant, Proc. London Math.Soc. 7 975-876, 208-22. [] M. Rosen, Number Theory in Function Field, GTM 20,New York Springer Verlag,2002 Received: September 2, 2008