<p>We study the structure of the Hilbert space of gauged matrix models with a global symmetry. In the first part of the paper, we focus on bosonic matrix models with U(2) gauge group and SO(<i>d</i>) global symmetry, and consider singlets under both the gauge and global symmetry. We show how such “double-gauged” matrix models can be described in terms of a simpler SO(3) single-matrix model. In the second part of the paper, we consider the so-called BMN subsector of the <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> <mo>=</mo> <mn>4</mn> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N}=4 \)</EquationSource> </InlineEquation> SU(<i>N</i>) super Yang-Mills theory, which is closely related to the BMN matrix model. Among the 1/16 BPS operators in this sector, “non-graviton” operators were recently discovered, which are expected to relate to the microstates of supersymmetric <i>AdS</i>₅ black holes. We show that a double gauging of this model, where one projects onto SU(3)_<i>R</i> R-symmetry singlets, considerably simplifies the analysis of the non-graviton spectrum. In particular, for low values of <i>N</i>, we show that (almost) all graviton operators project out of the spectrum, while important classes of non-graviton operators remain. In the <i>N</i> = 3 case, we obtain a closed form expression for the superconformal index of singlet non-gravitons, which reveals structural features of their spectrum.</p>