A matrix $A=(a_{nk})$ is called {\it normal\/} if $a_{nk}=0$ for $k>n$ and $a_{nn}\neq 0$ for all $n$. Such a matrix has a normal inverse $A^{-1}=(\alpha_{nk})$. If Ihe inverse $A^{-1}$ of a normal and regular matrix $A$ satisfies the conditions $\alpha_{nk}\leq 0$ for $k0$ for all $n$, we call such a matrix a Beekmann matrix. Beekmann introduced those matrices and proved that for such a matrix $A$, the matrix $B=(I+\lambda A)/(1+\lambda)$ is Mercerian for $\lambda>-1$. (I is the identity matrix.) This paper extends Beekmann's theorem to the case of $R_\beta$-Mercerian matrices, $\beta>0$.