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A data matrix may be seen simply as a means of organizing observations intorows ( e.g., by measured object) and into columns ( e.g., by measured variable)so that the observations can be analyzed with mathematical tools. As amathematical object, a matrix defines a linear mapping between pointsrepresenting weighted combinations of its rows (the row vector space) andpoints representing weighted combinations of its columns (the column vectorspace). From this perspective, a data matrix defines a relationship between theinformation that labels its rows and the information that labels its columns,and numerical methods are used to analyze this relationship. A first step is tonormalize the data, transforming each observation from scales convenient formeasurement to a common scale, on which addition and multiplication canmeaningfully combine the different observations. For example, z-transformationrescales every variable to the same scale, standardized variation from anexpected value, but ignores scale differences between measured objects. Here wedevelop the concepts and properties of projective decomposition, which appliesthe same normalization strategy to both rows and columns by separating thematrix into row- and column-scaling factors and a scale-normalized matrix. Weshow that different scalings of the same scale-normalized matrix form anequivalence class, and call the scale-normalized, canonical member of the classits scale-invariant form that preserves all pairwise relative ratios.Projective decomposition therefore provides a means of normalizing the broadclass of ratio-scale data, in which relative ratios are of primary interest,onto a common scale without altering the ratios of interest, and simultaneouslyaccounting for scale effects for both organizations of the matrix values. Bothof these properties distinguish it from z-transformation.

Projective Decomposition and Matrix Equivalence up to Scale