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This article gives a formal definition of a lognormal family of probabilitydistributions on the set of symmetric positive definite (PD) matrices, seen asa matrix-variate extension of the univariate lognormal family of distributions.Two forms of this distribution are obtained as the large sample limitingdistribution via the central limit theorem of two types of geometric averagesof i.i.d. PD matrices: the log-Euclidean average and the canonical geometricaverage. These averages correspond to two different geometries imposed on theset of PD matrices. The limiting distributions of these averages are used toprovide large-sample confidence regions for the corresponding population means.The methods are illustrated on a voxelwise analysis of diffusion tensor imagingdata, permitting a comparison between the various average types from the pointof view of their sampling variability.

Lognormal Distributions and Geometric Averages of Positive Definite Matrices