Projective Decomposition and Matrix Equivalence up to Scale

A data matrix may be seen simply as a means of organizing observations into rows ( e.g., by measured object) and into columns ( e.g., by measured variable) so that the observations can be analyzed with mathematical tools. As a mathematical object, a matrix defines a linear mapping between points representing weighted combinations of its rows (the row vector space) and points representing weighted combinations of its columns (the column vector space). From this perspective, a data matrix defines a relationship between the information that labels its rows and the information that labels its columns, and numerical methods are used to analyze this relationship. A first step is to normalize the data, transforming each observation from scales convenient for measurement to a common scale, on which addition and multiplication can meaningfully combine the different observations. For example, z-transformation rescales every variable to the same scale, standardized variation from an expected value, but ignores scale differences between measured objects. Here we develop the concepts and properties of projective decomposition, which applies the same normalization strategy to both rows and columns by separating the matrix into row- and column-scaling factors and a scale-normalized matrix. We show that different scalings of the same scale-normalized matrix form an equivalence class, and call the scale-normalized, canonical member of the class its scale-invariant form that preserves all pairwise relative ratios. Projective decomposition therefore provides a means of normalizing the broad class of ratio-scale data, in which relative ratios are of primary interest, onto a common scale without altering the ratios of interest, and simultaneously accounting for scale effects for both organizations of the matrix values. Both of these properties distinguish it from z-transformation.