  In ergodic quantum systems, physical observables have a non-relaxing
component if they "overlap" with a conserved quantity. In interacting
microscopic models, how to isolate the non-relaxing component is unclear. We
compute exact dynamical correlators governed by a Hamiltonian composed of two
large interacting random matrices, $H=A+B$. We analytically obtain the
late-time value of $\langle A(t) A(0) \rangle$; this quantifies the
non-relaxing part of the observable $A$. The relaxation to this value is
governed by a power-law determined by the spectrum of the Hamiltonian $H$,
independent of the observable $A$. For Gaussian matrices, we further compute
out-of-time-ordered-correlators (OTOCs) and find that the existence of a
non-relaxing part of $A$ leads to modifications of the late time values and
exponents. Our results follow from exact resummation of a diagrammatic
expansion and hyperoperator techniques.
