  Let $\mathcal{I},\mathcal{J}$ be two ideals on $\mathbf{N}$ which contain the
family $\mathrm{Fin}$ of finite sets. We provide necessary and sufficient
conditions on the entries of an infinite real matrix $A=(a_{n,k})$ which maps
$\mathcal{I}$-convergent bounded sequences into $\mathcal{J}$-convergent
bounded sequences and preserves the corresponding ideal limits. The well-known
characterization of regular matrices due to Silverman--Toeplitz corresponds to
the case $\mathcal{I}=\mathcal{J}=\mathrm{Fin}$.
  Lastly, we provide some applications to permutation and diagonal matrices,
which extend several known results in the literature.
