Selçuk J. Appl. Math. Selçuk Journal of Vol. 7. No.1. pp. 63-68, 2006 Applied Mathematics
On The Hadamard Exponential GCD-Matrices Ay¸se Nallı
Department of Mathematics, Selcuk University, 42031, Campus-Konya,Turkey; Received : August 25, 2005
Summary.In this paper we study the n×n Hadamard exponential GCD matrix = £()¤=1 whose -entry is ().We give the structure theorem and calculate the determinant, the trace, inverse and determine upper bound for determinant of the Hadamard exponential GCD matrix. Furtermore determine lower bound for the Euclidean norm of the Hadamard exponential GCD matrix. Key words: Hadamard exponential GCD matrix , GCD matrix , LCM matrix. 2000 Mathematics Subject Classification : 11C20, 11A05, 15A36, 11A25. 1. Introduction
Let = {1 2 } be an ordered set of distinct positive integers. The × matrix () whose -entry is the greatest common divisor ( ) of and is called the GCD matrix on S. The × matrix [] whose -entry is the least common multiple [ ] of and is called the LCM matrix. H. J. S. Smith [7] calculated the determinant of the GCD and LCM matrices. The results on GCD and LCM matrices have been presented in the literature. For general accounts see e.g. [2], [3] and [6].
In this paper we study basic properties of the n×n Hadamard exponential GCD matrix =£()¤
=1 whose -entry is
() on S.
We give the structure theorem and calculate the determinant, the trace, in-verse and determine upper bound for determinant of the Hadamard exponential GCD matrix. Furtermore determine lower bound for the Euclidean norm of the Hadamard exponential GCD matrix.
Throughout this paper let = {1 2 } be an ordered set of distinct positive integers and let = () be the × Hadamard exponential GCD matrix whose -entry is ()defined on S.
Note that the × Hadamard exponential GCD matrix is symmetric.
We present a structure theorem and then we calculate the value of the determi-nant of the × Hadamard exponential GCD matrix .The following theorem describes the structure of the × Hadamard exponential GCD matrix . Theorem 1. Let be the n×n Hadamard exponential GCD matrix whose -entry is ().Then
(1) = Γ
where is the × lower triangular matrix whose -entry is given by
(2) =
½
1 |
0
and Γ = ( (1) (2) () ) is the × diagonal matrix whose diagonal elements are
(3) () =X
|
( ) Proof. The -entry of Γ is equal to
( Γ )= X =1 () = X | | () = X |() ()
If we apply Möbius inversion formula to (3), then we obtainP
|
() = . Then
( Γ )
= P |()
() = () Thus, we obtain Theorem 1.
Corollary 1.Let () be as in (3). If is a prime then () = − If = 1 1 2 2 then () = 11 22 − 1−11 22 − 11 2−12 + 1−11 2−12
where 1 2are distinct primes.
Theorem 2. Let be the × Hadamard exponential GCD matrix on S. Then det = Y =1 ()
Proof. By Theorem 1, we can write = Γ . The × matrix
is a lower triangular matrix whose diagonal elements are (1 1 1) and Γ = ( (1) (2) () ) is the × diagonal matrix. Then
det = (det )(det Γ)(det ) =
Y
=1
()
Example 1. Consider the 5 × 5 Hadamard exponential GCD matrix
= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 2 2 3 2 4 5 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ det = 5 Y =1 () = (1)(2)(3)(4)(5) By using Corollary 1, we obtain
(1) = (2) = (2− ) (3) = (3− ) (4) = (4− 2) (5) = (5− )
= 6( − 1)4( + 1)3(2+ 1)
Corollary 2. Let be the × Hadamard exponential GCD matrix on S. Then
() =
+1−
− 1 Proof. The -entry of is equal to (). Then,
(()) = X =1 = +1 − − 1
Corollary 3. The n×n Hadamard exponential GCD matrix is positive definite. Proof : From [4], Corollary 3 is trivial.
Corollary 4. Let E be the × Hadamard exponential GCD matrix on S. Then
6 (+1)2
Theorem 3. Let be the × Hadamard exponential GCD matrix on S. Then E is invertible and
(()−1) = X []| 1 () ( )( )
Proof. From Corollary 3, we obtain det 0 then the n×n Hadamard expo-nential GCD matrix E is invertible.
Let be as in (2) and be defined as follow
=
(
³´ | 0 Calculating the -entry of the product gives
( ) = X =1 = X | | ( ) = X | () = ½ 1 = 0 6=
Thus −1= . From Theorem 1 we write = Γ . Therefore
−1 = (−1)Γ−1−1= Γ−1 (−1)= X =1 1 ()= X | | 1 () ( ) ( ) = X ]| 1 () ( ) ( )
Example 2. Consider the 5 × 5 Hadamard exponential GCD matrix = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 2 2 3 2 4 5 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ From Theorem 3 (−1)= = X []| 1 () ( ) ( ) 11 = 1 (1) (1) 2+ 1 (2) (2) 2+ 1 (3) (3) 2+ 1 (4)(4) 2+ 1 (5) (5) 2 = 4+ 3+ 22+ + 2 5− 12= 1 (2) (2) (1) + 1 (4) (4)(2) = −1 2− 13= 1 (3)(3)(1) = −1 3− 14= 1 (4)(4)(1) = 0 15= 1 (5)(5)(1) = −1 5− 22= 1 (2)(1) 2+ 1 (4)(2) 2=2+ + 1 4− 2 23= 0 24= 1 (4)(2)(1) = −1 4− 2 25= 0 33= 1 (3)(1) 2= 1 3− 34= 0 35= 0 44= 1 (4)(1) 2= 1 4− 2 45= 0 55= 1 (5)(1) 2= 1 5−
The × matrix −1= () is symmetric and
−1= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 4+3+22++2 5− −12− −13− 0 −15− −1 2− 2++1 4−2 0 4−1−2 0 −1 3− 0 1 3− 0 0 0 4−1−2 0 1 4−2 0 −1 5− 0 0 0 51− ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
Theorem 4. If be the × Hadamard exponential GCD matrix on S. Then the inequality p (1)(2()6√1 k k2
is valid for the Euclidean norm of the n×n Hadamard exponential GCD matrix. Proof. From 7 ,
()6 kk
2 2
where is the conjugate transpose of matrix A.
From Theorem 1, (det())26 kk 2 2 p (1)(2)()6 √1 kk2 Thus, the proof is completed.
References
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