On ANOVA-Like Matrix Decompositions

The analysis of variance plays a fundamental role in statistical theory and practice, the standard Euclidean geometric form being particularly well established. The geometry and associated linear algebra underlying such standard analysis of variance methods permit, essentially direct, generalisation to other settings. Specifically, as jointly developed here: (a) to minimum distance estimation problems associated with subsets of pairwise orthogonal subspaces; (b) to matrix, rather than vector, contexts; and (c) to general, not just standard Euclidean, inner products, and their induced distance functions. To make such generalisation, we solve the following problem: given a set of nontrivial subspaces of a linear space, any two of which meet only at its origin, exactly which inner products make these subspaces pairwise orthogonal? Applications in a variety of areas are highlighted, including: (i) the analysis of asymmetry, and (ii) asymptotic comparisons in Invariant Coordinate Selection and Independent Component Analysis. A variety of possible further generalisations and applications are noted.