A necessary and sufficient condition is given for a family of multivariate normal distributions with zero mean vectors to admit a complete sufficient statistic.
Justus Seely. "Completeness for a Family of Multivariate Normal Distributions." Ann. Math. Statist. 43 (5) 1644 - 1647, October, 1972. https://doi.org/10.1214/aoms/1177692396