Choosing the Best Exchange Vectors to Describe a Mineral with Solid Solutions

An algebraic method is developed for choosing exchange vectors that best describe composition variations in a mineral when several analyses are available. The method combines lower rank approximation of the matrix of chemical formulae with canonical angle analysis of proximity between spaces (both using singular value decomposition). After choosing the best exchange vectors, approximate mineral compositions are obtained. They can be expressed by their coordinates onto the exchange vectors taken as new axes or as approximate chemical formulae. The method is applied to a set of chemical analyses performed on several magnetite crystals with variable composition.