  We consider the family of arithmetical matrices given explicitly by
$$E(\sigma,\tau)=\left\{\frac{n^\sigma
m^\sigma}{[n,m]^\tau}\right\}_{n,m=1}^\infty,$$ where $[n,m]$ is the least
common multiple of $n$ and $m$ and the real parameters $\sigma$ and $\tau$
satisfy $\rho:=\tau-2\sigma>0$, $\tau-\sigma>\frac12$ and $\tau>0$. We prove
that $E(\sigma,\tau)$ is a compact self-adjoint positive definite operator on
$\ell^2({\mathbb N})$, and the ordered sequence of eigenvalues of
$E(\sigma,\tau)$ obeys the asymptotic relation
$$\lambda_n(E(\sigma,\tau))=\frac{\varkappa(\sigma,\tau)}{n^\rho}+o(n^{-\rho}),
\quad n\to\infty,$$ with some $\varkappa(\sigma,\tau)>0$. We give an
application of this fact to the asymptotics of singular values of truncated
multiplicative Toeplitz matrices with the symbol given by the Riemann zeta
function on the vertical line with abscissa $\sigma<1/2$. We also point out a
connection of the spectral analysis of $E(\sigma,\tau)$ to the theory of
generalised prime systems.
