  This article presents maximum likelihood estimators (MLEs) and log-likelihood
ratio (LLR) tests for the eigenvalues and eigenvectors of Gaussian random
symmetric matrices of arbitrary dimension, where the observations are
independent repeated samples from one or two populations. These inference
problems are relevant in the analysis of diffusion tensor imaging data and
polarized cosmic background radiation data, where the observations are,
respectively, $3\times3$ and $2\times2$ symmetric positive definite matrices.
The parameter sets involved in the inference problems for eigenvalues and
eigenvectors are subsets of Euclidean space that are either affine subspaces,
embedded submanifolds that are invariant under orthogonal transformations or
polyhedral convex cones. We show that for a class of sets that includes the
ones considered in this paper, the MLEs of the mean parameter do not depend on
the covariance parameters if and only if the covariance structure is
orthogonally invariant. Closed-form expressions for the MLEs and the associated
LLRs are derived for this covariance structure.
