comparing directions on the sphere

Hello everyone,

I am trying to compare two directions on the unit sphere, given two estimated mean directions a and b and their covariance matrices A and B. The covariance matrices do not have full rank.

The mean directions and covariance matrices are derived from the variance of entries of a 3x3 tensor and not from population samples.

Now I'd like some measure that tells me wether the angle acos(a'b) is "significant" given the covariance matrices A and B.

Maybe it's really simple, in this case sorry but I'm new to statistics :)

Thanks a lot!