Testing multinormality based on generalized inverse

Based on a characterization of multivariate normal distribution and the goodness-of-fit test for uniformity on the surface of a unit sphere, we present a new test for multivariate normal distribution. The new test possesses symmetry which improves on the combination tests based on the center of mass and the moment of inertia. The asymptotic chi-squared distribution of the test statistic is obtained, the test statistic has nice properties and it can be easily computed.