On the eigenvalues of certain number-theoretic matrices

In this paper we study the structure and give bounds for the eigenvalues of the $n\times n$ matrix, which $ij$ entry is $(i,j{)}^{\alpha }[i,j{]}^{\beta }$, where $\alpha ,\beta \in \text{Rset}$, $(i,j)$ is the greatest common divisor of $i$ and $j$ and $[i,j]$ is the least common multiple of $i$ and $j$. Currently only $O$-estimates for the greatest eigenvalue of this matrix can be found in the literature, and the asymptotic behaviour of the greatest and smallest eigenvalue is known in case when $\alpha =\beta$.