The mathematical extension from scalars to matrices as a prerequisite for univariate to multivariate generalizations of statistical concepts and methods is discussed. The basic notion of variability is generalized from a univariate context to a multivariate context using two matrix functions, a determinant and a trace. These two means of generalizing variance are shown to yield a number of alternative multivariate indices of shared variation. Multivariate partial indices are proposed. The complexity of multivariate test statistics and some problems in the interpretation of tests of multivariate hypotheses are reviewed.
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