A generalization of the ical finiteness theorem of Høegh-Krohn, Landstad and Størmer for ergodic actions of compact groups on operator algebras is proved for actions of compact matrix pseudogroups on $C^*$-algebras. This, together with the Takesaki-Takai type duality result of Baaj and Skandalis, show that the reduced $C^*$-crossed product of a unital $C^*$ - algebra by an ergodic action of a compact matrix pseudogroup is a direct sum of $C^*$-algebras of compact operators