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Generalized GCD matrices. (English) Zbl 1246.11077

The usual Smith type matrix is of the form [f((x i ,x j ))], where the ij entry is the arithmetical function f evaluated at the greatest common divisor (x i ,x j ) of x i and x j . The present author considers certain analogous matrices of the form [f(i,(i,j))]. As an open problem the author asks to evaluate the determinant of this matrix. Solutions to this problem in terms of general meet matrices on semilattices can be found in [B. Lindström, Proc. Am. Math. Soc. 20, 207–208 (1969; Zbl 0165.02902)] and in [M. Mattila and P. Haukkanen, Some properties of row-adjusted meet and join matrices, Linear Multilinear Algebra, in print].

For a general account on this type matrices, see [J. Sándor and B. Crstici, Handbook of number theory. II. Dordrecht: Kluwer Academic Publishers (2004; Zbl 1079.11001)].

MSC:
11C20Matrices, determinants (number theory)
11A25Arithmetic functions, etc.
15B36Matrices of integers