Symmetric positive semi-definite (PSD) matrices appear as data objects in the statistical analysis of Diffusion Tensor Imaging data, where there is interest in making inferences about the eigenvalues and eigenvectors of these objects. In this talk, I present a stratification of the set of symmetric PSD matrices of arbitrary dimension according to their eigenvalues, as well as maximum likelihood estimators (MLEs) and log-likelihood ratio (LLR) tests for the eigenvalues and eigenvectors of the mean matrix in a symmetric-matrix Gaussian model. The parameter sets involved are subsets of Euclidean space that are either affine subspaces, polyhedral convex cones, or orthogonally invariant embedded submanifolds. The asymptotic behavior of the MLEs and LLRs depend on the stratum where the true mean matrix lies.