On Fermat Power GCD Matrices Defined on Special Sets of Positive Integers

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1 Applied Mathematical Sciences, Vol 13, 2019, no 8, HIKARI Ltd, wwwm-hikaricom On Fermat Power GCD Matrices Defined on Special Sets of Positive Integers Yahia Awad, Ragheb Mghames Department of Mathematics and Physics, School of Arts and Sciences Lebanese International University, West Bekaa, Lebanon Haissam Chehade, Wiam Zeid Department of Mathematics and Physics, School of Arts and Sciences International University of Beirut, Saida, Lebanon This article is distributed under the Creative Commons by-nc-nd Attribution License Copyright c 2019 Hikari Ltd Abstract In this article, we define the n n Fermat Power GCD matrices defined on non factor-closed and non gcd-closed sets A complete characterization of their factorizations, determinants, reciprocals, and inverses are given Mathematics Subject Classification: 11A25, 15A09, 15A15, 15A23 Key words and phrases: Fermat Power GCD Matrix, factor-closed sets, gcd-closed sets, Fermat numbers 1 Introduction Let T {t 1, t 2,, t n } be a well ordered set of n distinct positive integers with t 1 < t 2 < < t n The GCD matrix T ) is a n n square matrix such that T ) ij t i, t j ), where t i, t j ) is the greatest common divisor of t i and t j The power GCD matrix on T is also n n square matrix such that T r ) ij t i, t j ) r, Corresponding author

2 370 Yahia Awad, Ragheb Mghames, Haissam Chehade and Wiam Zeid where r is any real number The set T {t 1, t 2,, t n } is said to be factorclosed if t k T for any divisor t k of t i T, and T is said to be gcd-closed if t i, t j ) T, for every t i and t j in T In 1876, Smith [15] showed that if T {t 1, t 2,, t n } is a factor-closed set of distinct positive integers, then dett ) φt 1 )φt 2 )φt n ) In 1996, S Z Chun [7] introduced the concept of the r th power GCD matrices for any real number r Moreover, he calculated the determinants of the r th power GCD matrices defined on both factor-closed and gcd-closed sets In addition, he obtained the structures for their inverses and reciprocals over the domain of natural numbers Now, let T {t 1, t 2,, t n } be a set of distinct positive integers Then, the Fermat power GCD matrix F r ) defined [ on T is the n n square matrix whose ij th entries are of the form f ij ) r 2 2t i,t j r, ) + 1] where r is any real number In this article, we define the n n Fermat Power GCD matrix as F r ), for any real number r, on both factor-closed and gcd-closed sets Then, we give a full description of its factorizations, determinants, reciprocals, and inverses 2 Fermat Power GCD Matrices Defined on Non Factor-Closed Sets In this section, we present structure theorems for Fermat power GCD matrices defined on arbitrary sets that are either factor-closed or not 21 Structure of Fermat Power GCD Matrices Definition 21 Let T {t 1, t 2,, t n } be a set of distinct positive integers, then the incidence matrix E n n e ij ) of T is the matrix defined as e ij 1 if t i t j and 0 otherwise Definition 22 Let T {t 1, t 2,, t n } be an arbitrary set of positive integers, the generalized Fermat power function on T is defined inductively for all 1 i n as gt i ) 2 2d + 1) r µt i /d), where r is any real number d t i Theorem 23 Let T {t 1, t 2,, t n } be a non factor-closed set of positive integers and T {y 1, y 2,, y m } be the factor-closed closure of T the minimal factor-closed set containing T ) Then, F r ) EA r E T, where E is the n m incidence matrix of T on T, and A r is the m m diagonal matrix such that a ii g y i )

3 On Fermat power GCD matrices defined on special sets of positive integers 371 Proof Since E is an n m incidence matrix of T on T, then e ij 1 if y j t i and 0 otherwise Hence, EA r E T ) ij e ik a kk e kj ) gy k ) y k t i,t j ) d t i,t j ) y k t i y k t j 2 2d + 1) r µ d y k 2 2d + 1) r y k t i,t j ) yk ) µ d d y k [ 2 2t i,t j ) + 1] r gy k ) yk ) 2 2d + 1) r d y k t i,t j ) Theorem 24 Let T {t 1, t 2,, t n } be a non factor-closed set of positive integers, and T {y 1, y 2,, y m } be the factor-closed closure of T Then, F r ) A r E T where a ij gy j ) if y j t i and 0 otherwise, and E is the corresponding incidence matrix relative to A r Proof The ij th entries of the incidence matrix E relative to A r are defined as: e ij 1 if a ij 0 and 0 otherwise So, the ij th entry of A r E T is Ar E ) T a ij ik e kj ) gy k ) y k t i y k t j y k t i,t j ) gy k ) [ 2 2t i,t j ) + 1] r Theorem 25 Let T {t 1, t 2,, t n } be a non factor-closed set of positive integers, and T {y 1, y 2,, y m } be the factor-closed closure of T Then, F r ) A r A T r where a ij gy j ) if y j t i and 0 otherwise Proof The ij th entry of A r A T r is defined as: A r A T r ) ij a ik a kj ) gyk ) gy k ) y k t i y k t j y k t i,t j ) gy k ) 22 Determinant of Fermat Power GCD Matrices [ 2 2t i,t j ) + 1] r Theorem 26 Let T {t 1, t 2,, t n } be a non factor-closed set of positive integers, and T {y 1, y 2,,y m } be the factor-closed closure of T with n m Let E k1,k 2,,k n) r be the sub-matrix consisting of the k1 th, k2 th,, kn th columns of E for some indices k i such that 1 k 1 < < k n m Then, det F r ) ) 2 n det E k1,k 2,,k n) r ) d ti2 2d + 1) r ti µ d 1 k 1 <<k n m

4 372 Yahia Awad, Ragheb Mghames, Haissam Chehade and Wiam Zeid Proof Let A r a ij ) m m be defined as a ij 2 2d + 1) r µ ) t id if yj t i d t i and 0 otherwise, and let E e ij ) be the corresponding incidence matrix relative to A r Since A r is a triangular matrix with a ii 2 2d + 1) r µ t id ) for d t i all 1 i m, then the ij th entry of A r can be written as a ij e ij 2 2d + d t i 1) r µ ) t id Define, for some indices ki such that 1 k 1 < < k n m, the matrices A and E k1,k 2,,kn) k 1,k 2 to be the sub-matrices consisting of,,kn) the k1 th, k2 th,, kn th columns of A and E, respectively Then, A k1,k 2,,kn) E D k1,k 2,,kn) A r, where D Ar is the n n diagonal sub-matrix of A r whose diagonal entries are d ii 2 2d + 1) r µ ) t id Therefore, detark1 ),k 2,,kn) d t ) i nπ dete ) k1,k 2 Applying Cauchy-Binet formula, we obtain,,kn) d ii det F r ) det A r E T ) 1 k 1 <k 2 <<k n m 1 k 1 <<k n m 1 k 1 <k 2 <<k n m ) ) T det A rk1,k 2 det E,,kn) k1,k 2,,kn) ti dete ) ) rk1 2 2d + 1) r µ det E,,kn) rk1,,kn) d d t i ) d ti2 2d + 1) r ti µ ) 2 det E rk1,k d 2,,kn) In the case where T {t 1, t 2,, t n } is a factor-closed set of distinct positive integers, we have the following corollary Corollary 27 If T {t 1, t 2,, t n } is a factor-closed set of distinct positive integers, then A is n n diagonal matrix with diagonal entries a ii g t i ) and E is also n n square incidence matrix relative to A, and hence det F r ) n gt i ) Proof By Theorem 1, we have det F r ) detea r E T ) deta r ) n gt i ) By Theorem 2, we have det F r ) det A r E ) T det A r ) gt i ) By Theorem 3, we have det F r ) det ) n n ) A r A T r gti )) gtk ) n gt i ) n ) T

5 On Fermat power GCD matrices defined on special sets of positive integers Reciprocal of Fermat Power GCD Matrices In this section, we present the reciprocals of Fermat power GCD matrices defined on arbitrary sets that are either factor-closed or not A complete characterization is also given Definition 28 Let T {t 1, t 2,, t n } be a set of distinct positive integers, then the n n matrix F r ) ij 1 1 f ij ) r ) r is called the reciprocal of 2 2t i,t j ) +1 Fermat power GCD matrix Definition 29 Let T {t 1, t 2,, t n } be an arbitrary set of positive integers, the generalized Reciprocal of Fermat power function on T is defined inductively for all 1 i n as: ht) ) r 1 µti /d), where r is a real d t i number 2 2d +1) Theorem 210 Let T {t 1, t 2,, t n } be a non factor-closed set of distinct positive integers Then, F r ) EA r E T, where A r diaght 1 ), ht 2 ),, ht n )) and E is an incidence matrix of T, such that e ij 1 if t j t i and 0 otherwise Proof Let T {y 1, y 2,, y m } be the factor closed closure of T Define the m m diagonal matrix whose diagonal entries are a ii h y i ) for all 1 i m Let E be the n m incidence matrix of T relative to T such that e ij 1 if y j t i and 0 otherwise Then, EA r E T ) ij ik a kk e jk ) e hy k ) y k t i y k t j 1 2 2d + 1) µ yk ) r d y k t i,t j ) d y k y k t i,t j ) hy k ) 1 [ 2 2t i,t j ) + 1) ] r 24 Inverse of Fermat Power GCD Matrices Definition 211 Let T {t 1, t 2,, t n } be a set of distinct positive integers The inverse of Fermat power GCD matrix is the square n n matrix F r ) 1 such that F r ) F r ) 1 I n Theorem 212 Let T {t 1, t 2,, t n } be a factor-closed set of distinct positive integers, and let E be the incidence matrix relative to T, such that e ij 1 if t j t i and 0 otherwise Then, the inverse of E is the matrix M T such that m ij µ t i t j ) if t j t i and 0 otherwise Moreover, F r ) 1 MA 1 r M T

6 374 Yahia Awad, Ragheb Mghames, Haissam Chehade and Wiam Zeid Proof Since T is factor-closed, then E is an n n square invertible matrix such that EM T ) ij e ik m kj ) µt) if t j t i { m kj ) t tk ti t i 1 if tj t t j i 0 otherwise 0 otherwise This implies that E 1 M T, and hence F r ) 1 EA r E T ) 1 E 1) T Ar ) 1 E) 1 MA 1 r M T 3 Fermat Power GCD Matrices Defined on Non gcd-closed Sets In this section, we study Fermat Power GCD matrices defined on non gcdclosed sets Full description of the their factorizations, determinants, reciprocals, and inverses are given 31 Structure of Fermat Power GCD Matrices We prove three different factorizations for Fermat power GCD matrices over non gcd-closed sets Theorem 31 Let T {t 1, t 2,, t n } be a gcd-closed set of distinct positive integers, and let g t k ) 2 2d + 1) r µ ) t kd Then, d t k gt k ) f ij ) r t k t i,t j ) t k t j t k tu tu<t j Proof It is clear that any set T of distinct positive integers is contained in a gcd-closed set Denote by T to be the minimal gcd-closed set containing T It is worthwise to observe that T usually contains considerably fewer elements than any factor-closed set containing T Also, it is clear that gt k ) is not representative and counted only once Then, the sum tk tj t k tu tu<t j Therefore, gt k ) g t k ) f ij ) r t k t i,t j ) t k t j t k tu tu<t j t k t i,t j ) t k t j, t k tu, tu<t j gt k ) g t k )

7 On Fermat power GCD matrices defined on special sets of positive integers 375 Theorem 32 Let T {t 1, t 2,, t n } be a non gcd-closed set of distinct positive integers, and T {y 1, y 2,, y m } be the minimal gcd-closed set containing T, then F r ) EA r E T, where E is the incidence matrix relative to T and A r is an m m diagonal matrix Proof Let T {y 1, y 2,, y m } be the minimal gcd-closed set containing T Define the m m diagonal matrix A r as follows: A r diag gd), d y 1 yu<y 1 gd),, d y 2 yu<y 2 gd) where gn) d n2 2d + 1) r µ n ) Let E be the incidence matrix of T on T d such that e ij 1 if y j t i and 0 otherwise Then, EA r E T ) ij gd) e ik a k e jk ) y k t i y k tj d y k Theorem 33 Let T {t 1, t 2,, t n } be a set of distinct positive integers, and T {y 1, y 2,, y m } be the minimal gcd-closed set containing T gd) if y j t i Then F r ) A r E T d y, where a ij k and e ij yu<y k { 0 otherwise 1 if aij 0 0 otherwise Proof Since A r and E are n m matrices, then A r E T ) ij y k t i y k t j gd) d y k a ik e jk ) Theorem 34 Let T {t 1, t 2,, t n } be a non gcd-closed set of distinct positive integers, and let T {y 1, y 2,, y m } be the minimal gcd-closed set con- gd) if y j t i taining T Then F r ) A r A T d y r, where A r ) ij k 0 otherwise

8 376 Yahia Awad, Ragheb Mghames, Haissam Chehade and Wiam Zeid Proof A r A T r ) ij g y k ) y k t i,t j ) ik a jk ) a gd) gd) y k t i y k tj d y k d y k y k t i,t j ) gd) d y k 32 Determinants of Fermat Power GCD Matrices Theorem 35 Let T {t 1, t 2,, t n } be a non gcd-closed set of positive integers, and let T {y 1, y 2,, y m } be the minimal gcd-closed set containing T with n < m If E k1,k 2,,k m) r is the sub-matrix consisting of the k1 th, k2 th,, km th columns of E for some indices k i such that 1 k 1 < k 2 < < k m n, then detf r ) 1 k 1 <k 2 <<k m n ) m 2 det Ek1,k 2,,k m) r gd) Proof Let A a ij ) and E e ij ) be its corresponding incidence matrix, where a ij gd) if y j t i and 0 otherwise But, A is a diagonal matrix whose diagonal entries are a ii gd) for all 1 i m, so the ij th entry of A may be written as e ij gd) and F r ) EF r E T Define, for some indices k i such that 1 k 1 < k 2 < < k m n, the matrices A k1,k 2,,km) and E k1,k 2 to be the sub-matrices consisting of k th,,km) 1, k2 th,, km th columns of A and E respectively, then A k1,k 2 E D,,km) k1,k 2,,km) r, where D r is the m m diagonal sub-matrix of A r whose diagonal elements are d ii gd) ) mπ Therefore, deta ) dete ) k1,k 2,,km) k 1,k 2 Applying Cauchy-,,km) d ii

9 On Fermat power GCD matrices defined on special sets of positive integers 377 Binet formula, we get det[f r ] det A r E T 1 k 1 <k 2 <<k m n 1 k 1 <<k m n 1 k 1 <k 2 <<k m n ) ) T det A k1,k 2 det E,,km) k1,k 2,,km) dete r ) m k1,,km) gd) m gd) det E rk1,k 2,,km) det E rk1,,km) ) 2 ) T Corollary 36 Let T {t 1, t 2,, t n } be a gcd-closed set of distinct positive integers, then det[f r ] detea r E T ) n d t m tu<tm d tu d t m,t u<t m,d t u gd) 33 Reciprocals of Fermat GCD Power Matrices Theorem 37 Let T {t 1, t 2,, t n } be a non gcd-closed set of distinct positive integers, and T {y 1, y 2,, y m } be the minimal gcd-closed set containing T, then F r ) EA r E T, where A r diag d y 1, d y u, y u<y 1 hd), d y 2, d y u, y u<y 2 hd),, hd), d y u, y u<y m such that hn) d n ) r 1 µ n 2 2d + 1) d ) Proof Let T {y 1, y 2,, y m } be the minimal gcd-closed set containing T

10 378 Yahia Awad, Ragheb Mghames, Haissam Chehade and Wiam Zeid Then, EA r E T ) ij e ik a k e jk ) hd) y k t i, y k tj d y k, d y u, y i <y k ) r 1 hy k ) F r) y k t i,t j) 2 2t i,t j) ij + 1) 34 Inverse of Fermat GCD Power Matrices Theorem 38 Let T {t 1, t 2,, t n } be a gcd-closed set of positive integers, then the inverse of F r ) is F r ) 1 such that ) ) F r ) 1 ij t µ k t ti µ k tj t i t k gd) d y k, d y u, y u<y k t j t k Proof Since T is gcd-closed, then E is an n n square invertible matrix such that E 1 M T Then, F r ) 1 ij References EA r E T ) 1 ij m 1 m ik t j t k m fkk r kj ) ) t µ k t ti µ k tj t i gd) tk d y k, d y u, y u<y k E 1) T Ar ) 1 E 1) M A r ) 1 M T [1] S Beslin and A N El-Kassar, GCD matrices and Smith s determinant Over UFD, Bull Number Theory Related Topics, ), [2] S Beslin and S Ligh, Greatest common divisor matrices, Linear Algebra and its Applications, ), [3] S Beslin, Reciprocal GCD matrices and LCM matrices, Fabonacci Quart, ), 71-74

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