A characterization of summability matrices that preserve the asymptotic equivalence of two sequences is given, where asymptotic equivalence is defined with respect to an ideal I of subsets of N. Pobyvanets (1980) gave necessary and sufficient conditions for a nonnegative summability matrix A to have the property that, for two nonnegative sequences x and y bounded away from 0, the ratio Ax/Ay tends to 1 whenever x/y tends to 1; an analogous characterization is given where the convergence of x/y is with respect to an ideal I as well as a new proof of Pobyvanets’ theorem. Similar extensions are given of theorems of Marouf and Li, in particular characterizations of summabilty matrices that map sequences x and y such that x/y is I-convergent to 1 to sequences with the property that the ratio of the sums of their tails or the ratio of the supremum of their tails also tend to 1 with respect to an ideal J. The main results of this paper are characterizations of summability matrices that preserve the asymptotic equivalence of two sequences, where the measures of asymptotic equivalence are defined with respect to an ideal of subsets of N. The work builds on, and generalizes, similar theorems characterizing matrices that preserve asymptotic equivalence when defined using the standard definition of sequential convergence. The starting point for this paper is a 1980 theorem of Pobyvanets [16], which gives necessary and sufficient conditions for a nonnegative summability matrix A to have the property that, for two nonnegative sequences x and y bounded away from 0, one has that Ax/Ay tends to 1 whenever x/y tends to 1. In this paper we establish this result (and others) where the convergence of x/y is with respect 2010 Mathematics Subject Classification. Primary 40A35; Secondary 40G15.