  We consider the family of arithmetical matrices given explicitly by
$$E=\left\{\frac{[n,m]^t}{(nm)^{(\rho+t)/2}}\right\}_{n,m=1}^\infty$$ where
$[n,m]$ is the least common multiple of $n$ and $m$ and the real parameters
$\rho$ and $t$ satisfy $t>0$, $\rho>t+1$. We prove that $E$ is a compact
self-adjoint operator on $\ell^2(\mathbb N)$ with infinitely many of both
positive and negative eigenvalues. Furthermore, we prove that the ordered
sequence of positive eigenvalues of $E$ obeys the asymptotic relation
$$\lambda^+_n(E)=\frac{\varkappa}{n^{\rho-t}}(1+o(1)), \quad n\to\infty,$$ with
some $\varkappa>0$ and the negative eigenvalues obey the same relation, with
the same asymptotic coefficient $\varkappa$. We also indicate a connection of
the spectral analysis of $E$ to the theory of Beurling primes.
