Matrix analysis of metamorphic mineral assemblages and reactions

Abstract

Assemblage diagrams are widely used in interpreting metamorphic mineral assemblages. In simple systems, they can help to identify assemblages which may represent equilibrium states; to determine whether differences between assemblages reflect changes in metamorphic grade or variations in bulk composition; and to characterize isograd reactions. In multicomponent assemblages these questions can be approached by investigating the rank, composition space (range) and reaction space (null-space) of a matrix representing the compositions of the phases involved. Singular value decomposition (SVD) provides an elegant way of (1) finding the rank of a matrix and detemining orthonormal bases for both the composition space and the reaction space needed to represent an assemblage or pair of assemblages; and (2) finding a model matrix of specified rank which is closest in a least squares sense to an observed assemblage. Although closely related to least squares techniques, the SVD approach has the advantages that it tolerates errors in all observations and is computationally simpler and more stable than non-linear least squares algorithms. Models of this sort can be used to interpret multicomponent mineral assemblages by straightforward generalizations of the methods used to interpret assemblage diagrams in simpler systems. SVD analysis of mineral assemblages described by Lang and Rice (1985) demonstrates the utility of the approach.