JP Journal of Algebra, Number Theory and Applications
Volume 3, Issue 2, Pages 245 - 257
(August 2003)
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ON
THE GCUD-RECIPROCAL LCUM MATRICES
Ayse Nalli (Turkey) and Dursun Tasci (Turkey)
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Abstract: Let
be an ordered set of distinct positive
integers. Then
matrix
having as
its ij-entry is called GCUD-Reciprocal
LCUM matrix on S, where
is the greatest common unitary divisor [Nederl.
Akad. Wetensch. Proc. Ser. A 64 (1961),
508-515] of
and
is the least common unitary multiple [Math.
Mag. 52 (1979), 217-222] of
and
A divisor d of x is said to be a
unitary divisor [Math. Zeit. 74 (1960), 66-80]
of x if
If d is a unitary divisor of x,
we write
We obtain a structure
theorem for the GCUD-Reciprocal LCUM matrices
defined on S. If S is unitary
divisor (ud)-closed, we calculate the
determinant of the GCUD-Reciprocal LCUM matrix
defined
on S and show that it is positive
definite. The set S is said to be
unitary divisor (ud)-closed [Linear and
Multilinear Algebra 41 (1996), 233-244] if all
unitary divisors of any member of S belong
to S. We obtain the trace and the value
of the determinant of the GCUD-Reciprocal LCUM
matrix defined on an arbitrary ordered set of
distinct positive integers. If S is ud-closed,
then we calculate the inverse of the GCUD-Reciprocal
LCUM matrix |
Keywords and phrases: unitary divisor, greatest common unitary
divisor, least common unitary multiple, GCUD-Reciprocal
LCUM matrix. |
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