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Abstract

Problem statement: Let T be a set of n distinct positive integers, x<SUB>1</SUB>, x<SUB>2</SUB>, ..., x<SUB>n</SUB>. The n�n matrix [T] having (x<SUB>i</SUB>, x<SUB>j</SUB>), the greatest common divisor of x<SUB>i</SUB> and x<SUB>j</SUB>, as its (i,j)-entry is called the Greatest Common Divisor (GCD) matrix on T. The matrix [[T]] whose (i,j)-entry is [x<SUB>i</SUB>, x<SUB>j</SUB>], the least common multiple of x<SUB>i</SUB> and x<SUB>j</SUB>, is called the least common multiple (LCM) matrix on T. Many aspects of arithmetics in the domain of natural integers can be carried out to Principal Ideal Domains (PID). In this study, we extend many recent results concerning GCD and LCM matrices defined on Factor Closed (FC) sets to an arbitrary PID such as the domain of Gaussian integers and the ring of polynomials over a finite field. Approach: In order to extend the various results, we modified the underlying computational procedures and number theoretic functions to the arbitrary PIDs. Properties of the modified functions and procedures were given in the new settings. Results: Modifications were used to extend the major results concerning GCD and LCM matrices defined on FC sets in PIDs. Examples in the domains of Gaussian integers and the ring of polynomials over a finite field were given to illustrate the new results. Conclusion: The extension of the GCD and LCM matrices to PIDs provided a lager class for such matrices. Many of the open problems can be investigated in the new settings.
Journal of Mathematics and Statistics 5 (4): 342-347, 2009
ISSN 1549-3644
© 2009 Science Publications
Corresponding Author: A.N. El-Kassar, Lebanese American University, P.O. Box 13-5053, Chouran-Beirut, 1102 2801,
Lebanon Tel: 961-1-786456, Ext. 1189 Fax: 961-1-867098
342
Greatest Common Divisor and Least Common Multiple Matrices on
Factor Closed Sets in a Principal Ideal Domain
1
A.N. El-Kassar,
2
S.S. Habre and
3
Y.A. Awad
1
Lebanese American University, P.O. Box 13-5053,
Chouran-Beirut, 1102 2801, Lebanon
2
Lebanese American University, Lebanon
3
Lebanese International University, Lebanon
Abstract: Problem statement: Let T be a set of n distinct positive integers, x
1
, x
2
, ..., x
n
. The n×n
matrix [T] having (x
i
, x
j
), the greatest common divisor of x
i
and x
j
, as its (i,j)-entry is called the
greatest common divisor (GCD) matrix on T. The matrix [[T]] whose (i,j)-entry is [x
i
, x
j
], the least
common multiple of x
i
and x
j
, is called the least common multiple (LCM) matrix on T. Many aspects
of arithmetics in the domain of natural integers can be carried out to Principal Ideal Domains (PID). In
this study, we extend many recent results concerning GCD and LCM matrices defined on Factor
Closed (FC) sets to an arbitrary PID such as the domain of Gaussian integers and the ring of
polynomials over a finite field. Approach: In order to extend the various results, we modified the
underlying computational procedures and number theoretic functions to the arbitrary PIDs. Properties
of the modified functions and procedures were given in the new settings. Results: Modifications were
used to extend the major results concerning GCD and LCM matrices defined on FC sets in PIDs.
Examples in the domains of Gaussian integers and the ring of polynomials over a finite field were given
to illustrate the new results. Conclusion: The extension of the GCD and LCM matrices to PIDs provided
a lager class for such matrices. Many of the open problems can be investigated in the new settings.
Key words: GCD matrix, lcm matrix, factor-closed sets, principal ideal domain
INTRODUCTION
Let T = {x
1
, x
2
, ..., x
n
} be a set of n distinct positive
integers. The n×n matrix [T] having (x
i
, x
j
), the greatest
common divisor of x
i
and x
j
, as its (i,j)-entry is called
the Greatest Common Divisor (GCD) matrix on T. The
matrix [[T]] whose (i,j)-entry is [x
i
,x
j
], the least
common multiple of x
i
and x
j
, is called the least
common multiple (LCM) matrix on T. The set T is said
to be factor closed (FC) if it contains every divisor of x
for any xT. In 1876, Smith
[11]
showed that the
determinant of the GCD matrix [T] on a FC set T is the
product
n
i
i 1
(x )
=
ϕ
, where ϕ is Euler's totient phi-
function. Moreover, Smith considered the determinant
of the LCM matrix on a FC set and showed that it is
equal to the product
n
i i
i 1
(x ) (x )
=
ϕ π
, where π is a
multiplicative function defined for a prime power p
r
by
π(p
r
) = p. Since then many papers related to Smith's
results have been published. Recently, this field has
been studied intensively. This new inspiration started in
by Beslin and Ligh
[3,4]
.
In
[3]
, Beslin and Ligh obtained a structure theorem
for GCD matrices and showed that, if S is FC, then
n
i
i 1
det[T] (x )
=
= ϕ
. They conjectured that the converse is
true. In
[10]
, Li proved the converse and provided a
formula for the determinant of an arbitrary GCD
matrix. Beslin and Ligh
[4,5]
generalized these results by
extending the FC sets to a larger class of sets, gcd-
closed sets. In
[1]
, a structure theorem for [[T]] was
obtained from the structure of the reciprocal GCD
matrix 1/[T], the (i,j)-entry of which is 1/(x
i
, x
j
). Given
a FC set T, Bourque and Ligh
[6]
calculated the inverses
of [T] and [[T]] and showed that [[T]][T]
1
is an
integral matrix. In that study, they stated their famous
conjecture that the LCM matrix on any gcd-closed set is
invertible. Bourque and Ligh
[7,8]
investigated the
structures, the determinants and the inverses associated
with classes of arithmetical functions. For a brief
J. Math. & Stat., 5 (4): 342-347, 2009
343
review of papers relating to Smith's determinant, we
refer to
[9]
. Using the language of posets, the authors
gave a common structure that is present in many
extensions of Smith’s determinants. Beslin and El-
Kassar
[2]
extended the results in
[3]
to unique
factorization domains.
The purpose of this study is to extend many of the
recent results concerning GCD and LCM matrices
defined on factor-closed sets to arbitrary Principal Ideal
Domains (PID) such as the domain of Gaussian integers
and the ring of polynomials over a finite field.
MATERIALS AND METHODS
Let S be a PID and let a, bS. We say that a and b are
associates and write a~b, if a = ub for some unit
element u in S. If b is a nonzero nonunit element, then b
has a unique factorization, up to associates, into prime
elements in S. That is,
1 2 i
1 2 i
α α α
=
, where the p
j
s
are distinct primes in S. Also, every finite set {b
1
, b
2
, ...,
b
n
} admits, up to associates, a greatest common divisor.
For a nonzero element b in S, define q(b) to be |S/<b>|,
the order of the quotient ring S/<b>, where <b> is the
principal ideal generated by b. Note that q(u) = 1, for
any unit u. Also note that in Z, Z[i] and Z
p
[x], q(b) is
finite b 0. Throughout the following we consider S
to be a PID having the property that q(b) is finite b
0. It can be shown that q(ab) = q(a)q(b). Hence, if
3
1 i
1 2 i
α
α α
=
, then
1 2 i
1 2 i
q(b) (q(p )) (q(p )) ...(q(p ))
α α α
=
.
For a positive integer n, q(n) = n and for a Gaussian
integer a+bi, q(a+bi) = a²+b². Also, if f(x) is a
polynomial of degree n in Z
p
[x], then q(f(x)) = p.
Define φ
s
(b) = |U(S/<b>)|, the order of the group of
units U(S/<b>). Then φ
s
(b) 1 and the equality holds
iff b is a unit. Also, φ
s
(b) = q(p)
1
iff p is prime in S. It
can be shown that φ
s
(b) is multiplicative and
if
3
1 i
1 2 i
α
α α
=
, then:
1
2 i
1
S 1 1 2
1 1
2 i i
(b) (q(p ) 1)(q(p )) (q(p ) 1)
(q(p )) ...(q(p ) 1)(q(p ))
α −
α α −
ϕ =
(1)
Now, if S = Z, then φ
s
(b) becomes Euler's phi-
function. Also, if
1 2 i
1 2 i
f(x) (p (x)) (p (x)) ...(p (x))
α α α
=
is a
polynomial of degree n in S = Z
p
[x], a product of
powers of distinct irreducible polynomials p
j
(x), 1 j
i, then:
ik k
k 1 k
i
r (
α
-1) r
sk 1
(f(x))=p (p 1)
=
=
ϕ −
(2)
where, p
j
(x) is of degree r
j
. For example, if f(x) = (x²+
1)
4
(x+1)
4
(x
3
+x
2
+x+4) in S = Z
2
[x], then φ
s
(f(x)) = 3
7
.
2
5
.13. Now, if
1 2 i
k k k
1 2 i
u ...
β = β β β
, a product of distinct
Gaussian primes β
j
= a
j
+ib
j
, 1 j i, then:
( ) ( )
j
ik 1
2 2 2 2
s j j j j
j 1
( )= a + b a + b 1
=
φ β
For example, if β = 6+42i ~ 3(1+i)
3
(1+2i)
2
, then
φ
s
(β) = 640.
Let S be a PID and let T = {t
1
, t
2
, ..., t
n
} be a set of
nonzero nonassociate elements in S. Define a linear
ordering on T according to the following scheme: If
q(t
i
) < q(t
j
) then t
i
t
j
and if the equality q(t
i
) = q(t
j
) holds
then order t
i
and t
j
according to any scheme depending
on the given domain S. For instance, if S = Z[i] and
q(t
i
) = q(t
j
), where t
i
~ a+ib, t
j
~ c+id, a, b, c, d 0, then
define t
i
t
j
whenever b < d. In the case S = Z
p
[i] and
q(t
i
(x)) = q(t
i
(x)), where t
i
(x) ~x+a
n1
x
1
+...+a
1
x+a
0
,
t
j
(x) ~ x+b
n1
x
1
+...+b
1
x+b
0
, 0 a
j
, b
j
p-1, j
0
is the
smallest index j such that a
j
b
j
, then define t
i
(x) t
j
(x)
whenever a
j0
<b
j0
. If the set T is ordered so that
t
1
t
2
...t
n
, we say that T is q-ordered. Two sets T and T
in S are associates, denoted by T~ T, iff each element
in T is associate to an element in T and vice versa. For
a nonzero element b, let E(b) be a complete set of
distinct nonassociate divisors of b in S. Then,
E(a)E(b) ~E((a, b)) and E(p
m
) ~{1, p
1
, p
2
,..., p
m
}. Note
that if t~t, then q(t) = q(t) and φ
s
(t) = φ
s
(t). Also,
S S
d T d T'
(d) (d)
∈ ∈
φ = φ
∑ ∑
whenever T~ T.
Theorem 1: Let S be a PID and let b be a nonzero
element in S. If E(b) is a complete set of distinct
nonassociate divisors of b, then
S
d E(b)
q(b) (d)
= φ
.
Proof: Let b 0. The result is true when b is a unit.
Suppose that b is a nonunit so that
1 2 i
1 2 i
α α α
=
.
Since φ
s
is multiplicative, the function
S
d E( b)
f(b) (d)
= ϕ
is also multiplicative. For any prime element p
j
, (1)
gives:
n
j
n 0 1 n
j S S j S j S j
d E(p )
j j j j
n 1
j
n n
j j
f(p ) (d) (p ) (p ) ... (p )
1 (q(p ) 1) (q(p ) 1)(q(p )) ... (q(p ) 1)
(q(p ))
(q(p )) q(p )
= ϕ = ϕ + ϕ + + ϕ
= + − + − + + −
= =
J. Math. & Stat., 5 (4): 342-347, 2009
344
By the multiplicativity of f(b), we have
S
d E(b)
q(b) (d)
= ϕ
.
Corollary 1: (Euler's) If n is a positive integer, then
d 0,d|n
n (d)
>
= ϕ
.
RESULTS
GCD matrices on FC sets in a PID: Throughout the
following, we consider T = {t
1
, t
2
, ..., t
n
} to be a q-
ordered set of nonzero nonassociate elements of a PID
S Define the GCD matrix on S to be the n×n matrix
[T] = (t
ij
) = q((t
i
, t
j
)). The set T is said to be a factor-
closed (FC) in S iff t
i
T and d|t
i
implies that d~ t
j
for
some t
j
T. Note that any set T in S is either factor-
closed or it is contained in a factor-closed set D.
Theorem 2: The GCD matrix [T] can be decomposed
into a product of an n×m matrix A and an m×n matrix
B, for some m n. The nonzero entries of A are equal
to φ
s
(d) for some d in a FC set D containing T and B is
an incidence matrix.
Proof: Let D = {d
1
, d
2
, ..., d
m
} be a FC set containing T
in the PID S Define the n×m matrix A = (a
ij
) by
s j j i
ij
(d ) if d E(t )
a
0 otherwise
ϕ ∈
=
and let B = (b
ij
) be the incidence
matrix corresponding to the transpose of A, where
ji
ij ji
1 if a 0
b
0 if a 0
=
=
. Hence, the product AB is given by:
k i k i j
k j
k i j
n
ij ik kj s k s k
k 1 d E (t ) d E(t ) E( t )
d E(t )
s k i j
d E((t ,t ))
(AB) = a b = (d ) (d )
(d ) q((t ,t ))
= ∈ ∩
ϕ = ϕ =
ϕ =
∑ ∑
Example 1: Let T = {1,1+x,1+x
3
,(1+x)
3
} in Z
2
[x].
Then [T] =
1 1 1 1
1 2 2 2
1 2 8 2
1 2 2 8
 
 
 
 
 
 
. Let D = {1, 1+x, (1+x)
2
,
1+x+x
2
, 1+x
3
, (1+x)
3
}. Then, [T]
4×4
= A
4×6
B
6×4
where:
A =
1 0 0 0 0 0
1 1 0 0 0 0
1 1 0 3 3 0
1 1 2 0 0 4
 
 
 
 
 
 
and B =
1 1 1 1
0 1 1 1
0 0 0 1
0 0 1 0
0 0 1 0
0 0 0 1
 
 
 
 
 
 
 
 
 
 
Example 2: Let T = {1, 2, 5}. In Z[i],
1 1 1
[T] 1 4 1
1 1 25
 
 
=
 
 
 
.
Note that T is not FC in Z[i]. Select D = {1, 1+i, 2, 2+i,
1+2i, 5}. Then
1 0 0 0 0 0
A 1 1 2 0 0 0
1 0 0 4 4 16
 
 
=
 
 
 
and
111
0 1 0
0 1 0
B
0 0 1
0 0 1
0 0 1
 
 
 
 
=
 
 
 
 
 
 
.
Theorem 3: The GCD matrix [T] is the product of an
n×m matrix A and its transpose A
T
. The nonzero entries
of A are of the form
S
(d)
ϕ for some d in a FC set D
containing T.
Proof: Let D = {d
1
, d
2
, ..., d
m
} be a FC set containing
T. Define the n×m matrix A = (a
ij
) by
s j j i
ij
(d ) if d E(t )
a
0 otherwise
ϕ ∈
=
. Hence, the product A A
T
is
given by:
k i
k j
k i j k i j
n
Tij ik kj s k s k
k 1 d E (t )
d E(t )
s k s k i j
d E(t ) E( t ) d E((t ,t ))
(AA ) = a b = (d ) (d )
= (d ) = (d ) = q((t ,t ))
= ∈
∈ ∩
ϕ ϕ
ϕ ϕ
∑ ∑
∑ ∑
Note that Theorem 2 and 3 hold even if T is not q-
ordered. In the case when both T and D are q-ordered,
B becomes in raw-echelon form.
Corollary 2: (Smith's Determinant over a PID) If T is
FC in S, then
[ ]
n
S i
i 1
det T (t ).
=
= ϕ
Proof: Let T be a FC set. Choose D to be q-ordered and
D ~ T. From Theorem 2, the GCD matrix [T] = AB,
where A is an n×n lower triangular matrix and B is an
upper triangular matrix such that a
ii
= φ
s
(t
i
) and b
ii
= 1, 1
i n. Therefore, det[T] = det[AB] = det[A]det[B] =
S 1 S 2 S n
(t ) (t )... (t )
ϕ ϕ ϕ
.
We note that if T =
1 n
i i2 i
{t ,t ,...,t }
is any
arrangement of the elements of T = {t
1
, t
2
, ..., t
n
} in S,
then det[T] = det[T]. This can be verified as follows.
J. Math. & Stat., 5 (4): 342-347, 2009
345
The matrix [T] can be obtained from [T] by switching
the rows and the columns of [T]. Thus, [T] = E
1
E
2
...
E
i
[T], where the E
j
s are elementary matrices with
det[E
j
] = ±1, 1 j i. Hence, [T] and [T] are similar
matrices and det[T] = det[T].
Next, we consider the converse of Corollary 2. Let
S be a PID and let T = {t
1
, t
2
, ..., t
n
} be a nonempty set
of nonzero nonassociate elements in S with
[ ]
n
S i
i 1
det T (t ).
=
= ϕ
Is it true that T is factor-closed in S?
Consider a minimal FC set D = {t
1
, t
2
, ..., t
n
, t
n+1
, ...,
t
n+r
} containing T = { t
1
, t
2
, ..., t
n
} with t
1
t
2
... t
n
and
t
n+1
t
n+2
... t
n+r
. Define an n×(n+r) matrix A by
(A)
ij
= ε
ij
S j
(t )
ϕ
, where ε
ij
is 1 if t
j
E(t
i
) and 0
otherwise. Denote the matrix (ε
ij
)
n×(n+r)
by E, a {0,1}-
matrix. Note that the matrix A is the same matrix A
defined in Theorem 3.
For an n×m matrix M, n > m and any set of indices
k
1
, k
2
, ..., k
n
with 1 k
1
< k
2
< ... < k
n
m, let
1 2 n
(k ,k ,..., k )
M
denote the submatrix consisting of k
1th
, k
2th
,
... k
nth
columns of M.
Theorem 4: Let D = {t
1
, t
2
,...,t
n
, t
n+1
, ..., t
n+s
} be a
minimal FC set containing T = { t
1
, t
2
,...,t
n
} in S, where
t
1
t
2
... t
n
and t
n+1
t
n+2
... t
n+s
. Then:
(
)
1 2 n 1 2 n
1 2 n
2
(k ,k ,...,k ) s k s k s k
1 k k ... k n s
det[T] det[E ] (t ) (t ).. (t )
< < ≤ +
= ϕ ϕ ϕ
Proof: Since [T] = AA
T
, Cauchy-Binet formula gives
that:
]
[
( )
( )
1 2 n 1 2 n
1 2 n
1 2 n
1 2 n
T
T
(k ,k ,...,k ) (k ,k ,...,k )
1 k k ... k n s
2
(k ,k ,...,k )
1 k k ... k n s
det T = det AA
det[A ]det[A ]
det[A ]
< < ≤ +
< < ≤ +
 
 
=
=
The result follows from the fact that:
1 2 n 1 2 n 1 2 n
(k ,k ,...,k ) (k ,k ,...,k ) s k s k s k
det[A det[E ] (t ) (t )... (t )
= ϕ ϕ ϕ
Corollary 3: Let [T] be the GCD matrix defined on T
in S. Then, det[T]
s 1 s 2 s n
(t ) (t )... (t )
ϕ ϕ ϕ .
Proof: The terms in the summation of Theorem 4 are
nonnegative. Since the submatrix
1 2 n
(k ,k ,...,k )
E is lower
triangular with diagonal elements equal to 1, we have
that the term corresponding to (k
1
, k
2
, ..., k
n
) = (1, 2, ...,
n) is det[E(1, 2, ..., n)]
2
φ
s
(t
1
) φ
s
(t
2
)... φ
s
(t
n
) = φ
s
(t
1
)
φ
s
(t
2
)... φ
s
(t
n
). Therefore, det[T]
s 1 s 2 s n
(t ) (t )... (t )
ϕ ϕ ϕ .
Theorem 5: Let [T] be the GCD matrix defined on T in
S. Then, det[T] =
s 1 s 2 s n
(t ) (t )... (t )
ϕ ϕ ϕ if and only if T is
factor-closed in S.
Proof: The sufficient condition holds from Corollary 2.
Conversely, suppose that det[T] =
s 1 s 2 s n
(t ) (t )... (t )
ϕ ϕ ϕ .
For contradiction purposes, suppose that T is not FC.
Let D = {t
1
, t
2
, ..., t
n
, t
n+1
, ..., t
n+s
} be a minimal FC set
containing T in S such that t
1
t
2
... t
n
and t
n+1
t
n+2
...
t
n+s
. Since T is not FC, D is not associate to T in S.
Then, t
n+1
is in D but not in T and t
n+1
E(t) for some t
in T. Now, let t
r
be the first element in T such that
t
n+1
E(t
r
). Then, the submatrix
(1,2,...r 1,n 1, r 1,...,n )
A
− + +
consisting of the 1
st
, 2
nd
, ..., (r
1
)
th
, (n+1)
th
, (r+1)
th
, ...
and n
th
columns of A
n×(n+s)
is a lower triangular
matrix of nonzero determinant. Hence,
(1,2,...r 1,n 1, r 1,...,n )
E
− + +
is a {0, 1}-matrix whose diagonal
elements are equal to 1. Since
(1,2,...r 1,r 1,..., n, n 1)
E
− + +
can be
obtained from
(1,2,...r 1,n 1, r 1,...,n )
E
− + +
by performing a certain
numbers of successive column permutations,
(1,2,...r 1,r 1,...,n ,n 1)
det[E ]
− + +
= ±
(1,2,...r 1,n 1,r 1,...,n)
det[E ]
+ +
= ±1.
From Theorem 4, we have:
(
)
1 2 n 1 2 n
1 2 n
2
(k ,k ,...,k ) s k s k s k
1 k k ... k n s
s 1 s 2 s n s 1 s 2 s r 1 s r 1
s n s n 1 s 1 s 2 s n
det[T] det[E ] (t ) (t )... (t )
(t ) (t )... (t ) (t ) (t )... (t ) (t )
... (t ) (t ) ... (t ) (t )... (t )
< < ≤ +
− +
+
= ϕ ϕ ϕ
= ϕ ϕ ϕ + ϕ ϕ ϕ ϕ
ϕ ϕ + > ϕ ϕ ϕ
This contradicts the necessary condition that the
equality holds.
Inverses of GCD matrices in a PID: Let t be any
nonzero element in S. The generalized Mobius function
over S is defined by:
m
s
1 if tisa unit
(t) (-1) if tis the product of m nonassocia
te primes
0 otherwise
µ =
Note that:
s
d E( t)
1 if tisa unit
(d)
0 otherwise
µ =
(3)
Corollary 4: Let [T] be the GCD matrix defined on T
in S. Then, [T] is invertible and its inverse [T]
1
= (r
ij
) is
given by:
J. Math. & Stat., 5 (4): 342-347, 2009
346
i k
j k
ij s k i s k j
t E (t ) s k
t E(t )
1
r (t / t ) (t / t )
(t )
= µ µ
ϕ
Proof: Define the n×n matrices E = (e
ij
) and
U = (u
ij
) as follows:
i j
ij
1 if t E(t )
e
0 otherwise
=
and
s i j i j
ij
(t / t ) if t E(t )
u
0 otherwise
µ ∈
=
. Then,
n
ij ik kj
k 1
(EU) e u
=
= =
j k k i j
i j
s i j s k
t E (t ) t E (t /t )
1 if t ~t
(t / t ) (t )
0 otherwise
∈ ∈
µ = µ =
∑ ∑
. The last
equality follows from (3). Since the elements in T are
nonassociate, we have U = E
1
. If D is the diagonal
matrix diag(φ
s
(t
1
), φ
s
(t
2
), ..., φ
s
(t
n
)) and A = ED
1/2
,
then [T] = AA
T
= (ED
1/2
)(ED
1/2
)
T
= EDE
T
. Therefore,
[T]
1
= U
T
D
1
U = (r
ij
), where r
ij
= (U
T
D
1
U)
ij
=
i k
j k
n
ki kj s k i s k j
k 1 t E (t )
s k s k
t E(t )
1 1
u u (t / t ) (t / t )
(t ) (t )
= ∈
= µ µ
ϕ ϕ
∑ ∑
.
Example 3: Let S = Z
2
[x] and let T = {1,1+x,1+x
2
,
(1+x)
3
,1+x+x
2
}, which is a q-ordered FC set of nonzero
nonassociate elements in Z
2
[x]. Then,
11111
1 2 2 2 1
[T]
1 2 4 4 1
1 2 4 8 1
1 1 1 1 4
 
 
 
 
=
 
 
 
 
. By corollary 4, [T]
1
is obtained
as follows:
11 2
S S S
1 1 1 1 7
a 1 1
(1) (1 x) (1 x x ) 3 3
= + + = + + =
ϕ ϕ + ϕ + +
12 S
1
a 1
(1 x)
= = −
ϕ +
13 14 12 2
S
1 1
a a 0, a
(1 x x ) 3
− −
= = = =
ϕ + +
and so forth. Therefore:
1
7 / 3 1 0 0 1 / 3
1 3 / 2 1 / 2 0 0
[T]
0 1/ 2 3 / 4 1 / 4 0
0 0 1/ 4 1 / 4 0
1/ 3 0 0 0 1/ 3
− −
 
 
− −
 
 
=− −
 
 
 
 
.
Reciprocal GCD Matrices in a PID: The reciprocal
GCD matrix on T in S is the n×n matrix 1/[T] whose
(i,j)-entry is 1/q((t
i
, t
j
)). It is clear that 1/[T] is
symmetric. Furthermore, permutations of the elements
of T yield similar reciprocal GCD matrices. For a
nonzero element t in T, define the function ξ
by
s
d E( t)
1
(t) q(d) (d)
q(t)
ξ = µ
. A generalized version of the
Mobius inversion formula can be used to show that
d E( t)
1
(d)
q(t)
= ξ
. Since ξ(t) is the product of two
multiplicative functions
1
q(t)
and
s
d E( t)
(t) q(d) (d)
χ = µ
,
we have that ξ(t) is itself multiplicative. Moreover, if p
is prime in S, then χ(p
n
) = 1q(p). Hence,
n
n
1 q(p)
(p )
(q(p))
ξ =
. Therefore:
S2
p E( t) p E(t )
1 (t)
(t) (1 q(p)) ( q(p))
q(t) (q(t))
∈ ∈
ϕ
ξ = =
∏ ∏
(4)
where, the product runs over all prime divisors p of t in
E(t).
In the following two theorems we obtain two
factorizations for the reciprocal GCD matrices.
Theorem 6: Let D = {d
1
, d
2
, ..., d
m
} be a FC set
containing T in S. The reciprocal GCD matrix defined
on T is the product of an n×m matrix A = (a
ij
), defined
by
j j i
ij
(d ) if d E(t )
a
0 otherwise
ξ ∈
=
and an m×n incidence matrix
B corresponding to A
T
.
Proof: Let A be as defined and let B be the m×n matrix
with
ij
ij ij
1 if a 0
b
0 if a 0
=
=
. Then:
j i j i j
j j
j i j
m
ij ik kj j j
k 1 d E(t ) d E(t ) E( t )
d E(t )
j
d E ((t ,t )) i j
(AB) a b (d ) (d )
1
(d ) q((t ,t ))
= ∈ ∩
= = ξ = ξ
= ξ =
∑ ∑
In a similar manner we prove the second
factorization given in the following theorem.
Theorem 7: Let D = {d
1
, d
2
, ..., d
m
} be a FC set
containing T in S and let C be the the n×n matrix given
by
j j i
ij
(d ) if d E(t )
a
0 otherwise
ξ ∈
=
. Then 1/[T] = CC
T
.
The proof of the following theorem is similar to
those of Theorems 4 and 5.
J. Math. & Stat., 5 (4): 342-347, 2009
347
Theorem 8: Let T be a set in S. Then, det(1/[T]) =
ξ(t
1
)ξ(t
2
)...ξ(t
n
) iff T is factor-closed in S.
LCM Matrices on FC Sets in a PID: The least
common multiple (LCM) matrix defined on T in S is
the n×n matrix [[T]] = (t
ij
), where t
ij
= q([t
i
, t
j
]) and [t
i
,
t
j
] is the least common multiple of t
i
and t
j
in S.
Theorem 9: If T is FC in S, then
[ ]
i
n
S i
i 1 p E(t )
det T (t ) ( q(p))
= ∈
 
= ϕ
 
 
 
 
∏ ∏
.
Proof: Since [t
i
, t
j
]~(t
i
t
j
)/( t,
i
t
j
), we have and q([t
i
,
t
j
]) = q(t
i
)q(t
j
)/q(( t,
i
t
j
)). Now q(t
i
) can be factored out
from the i
th
row and q(t
j
) from the j
th
column to obtain
1/[T]. Hence, [[T]] = D.(1/[T]).D, where D is the n×n
diagonal matrix with diagonal entries q(t
1
), q(t
2
), ...,
q(t
n
). From 4, we have that:
det[[S]] = det(D.(1/[T]).D) = det(D)²det(1/[T])
= (q(t
1
))²(q(t
2
))²...(q(t
n
))²ξ(t
1
)ξ(t
2
)...ξ(t
n
)
=
i
n
S i
i 1 p E(t )
(t ) ( q(p))
= ∈
 
ϕ −
 
 
 
∏ ∏
Cauchy Binet formula yields a formula for the
determinant of the LCM matrix defined on a set T
which is not necessarily FC. The formula is given by:
( )
( )
1 2 n
1 2 n
1 2 n 1 2 n
2
(k ,k ,...,k )
1 k k ... k n s
2
k k k k k k
det[[T]] det E
q(t )q(t )...q(t ) (t ) (t )... (t )
< < ≤ +
=
ξ ξ ξ
From Theorem 9, we have that the determinant of
the GCD matrix on S divides the determinant of the
LCM matrix whenever T is FC in S.
DISCUSSION
Most of the existing results related to GCD and
LCM matrices are obtained in the domain of natural
integers. The results are based on certain number
theoretic functions such as Euler’s phi function and the
Mobius function. These function and their properties
can be generalized to principal ideal domains. By
describing the underlying computational procedures
and the various properties in the new settings, the
existing results related to GCD and LCM defined on
factor closed sets are extended to PIDs. This provides a
large class of such matrices where many new examples
can be constructed. In particular, examples in the
domains of Gaussian integers and the ring of
polynomials over a finite field may give new insight to
some open problems.
CONCLUSION
The extension of the GCD and LCM matrices
to PIDs provide a lager class for such matrices. Many
of the open problems can be investigated in the new
settings. For future study, we suggest the problem of
extending GCD and LCM matrices defined on gcd
closed sets to PIDS.
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Article
In this study, we extend many recent results concerning GCD and LCM matrices defined on factor closed (FC) sets to an arbitrary PID such as the domain of Gaussian integers and the ring of polynomials over a finite field.