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Let $\mathcal{I},\mathcal{J}$ be two ideals on $\mathbf{N}$ which contain thefamily $\mathrm{Fin}$ of finite sets. We provide necessary and sufficientconditions on the entries of an infinite real matrix $A=(a_{n,k})$ which maps$\mathcal{I}$-convergent bounded sequences into $\mathcal{J}$-convergentbounded sequences and preserves the corresponding ideal limits. The well-knowncharacterization of regular matrices due to Silverman--Toeplitz corresponds tothe 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.

A characterization of $(\mathcal{I}, \mathcal{J})$-regular matrices