The family of arithmetical matrices is studied given explicitly by
E
(
σ
,
τ
)
=
{
n
σ
m
σ
[
n
,
m
]
τ
}
n
,
m
=
1
∞
,
\begin{equation*} E(\sigma ,\tau )= \bigg \{\frac {n^\sigma m^\sigma }{[n,m]^\tau }\bigg \}_{n,m=1}^\infty , \end{equation*}
where
[
n
,
m
]
[n,m]
is the least common multiple of
n
n
and
m
m
and the real parameters
σ
\sigma
and
τ
\tau
satisfy
ρ
≔
τ
−
2
σ
>
0
\rho ≔\tau -2\sigma >0
,
τ
−
σ
>
1
2
\tau -\sigma >\frac 12
, and
τ
>
0
\tau >0
. It is proved that
E
(
σ
,
τ
)
E(\sigma ,\tau )
is a compact selfadjoint positive definite operator on
ℓ
2
(
N
)
\ell ^2(\mathbb {N})
, and the ordered sequence of eigenvalues of
E
(
σ
,
τ
)
E(\sigma ,\tau )
obeys the asymptotic relation
λ
n
(
E
(
σ
,
τ
)
)
=
ϰ
(
σ
,
τ
)
n
ρ
+
o
(
n
−
ρ
)
,
n
→
∞
,
\begin{equation*} \lambda _n(E(\sigma ,\tau ))=\frac {\varkappa (\sigma ,\tau )}{n^\rho }+o(n^{-\rho }), \quad n\to \infty , \end{equation*}
with some
ϰ
(
σ
,
τ
)
>
0
\varkappa (\sigma ,\tau )>0
. This fact is applied 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
σ
>
1
/
2
\sigma >1/2
. The relationship of the spectral analysis of
E
(
σ
,
τ
)
E(\sigma ,\tau )
with the theory of generalized prime systems is also pointed out.