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Fisher (1989) advocated using the Singular Value Decomposition (SVD) of the composition matrix of a set of minerals for the determination of reactions between the minerals, and therefore of the compatibility relationships between mineral assemblages. The advantage of using the SVD for this is that an independent set of reactions, the left nullspace of the composition matrix, can be determined for a rank-deficient composition matrix which is close to, or within error of, the observed matrix. The zero singular values in the SVD can be identified by propagating the measurement uncertainties through to the singular values, and saying that singular values are zero if they are within error of zero. Having obtained an independent set of reactions, the reactions in the independent set can be combined linearly to find the full set of reactions. Propagating the measurement uncertainties through to the left nullspace of the composition matrix, the uncertainties in the coefficients of the reactions in the independent set can be determined. From these uncertainties, the uncertainties in all of the reactions can be determined, allowing the identification of reactions which are degenerate within error. This is important because it allows the identification of ambiguities in the compatibility relationships between mineral assemblages.
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