Quadratic constraints mineral reactions

Let us consider the problem of finding the stoichiometric coefficients of a mineral reaction, m element concentrations have been measured on n mineral phases of a rock (C/, i= 1. m j= 1. n) and it is suspected that the phases are not chemically independent. In other words, we can find n numbers Vj (/ = 1. n) such that 

Obviously, the trivial solution v,=0 (/ = L n) does not fit our needs and we must search for solutions as a constrained problem in which the solution vector is of constant, yet arbitrary, length. In other words, we become interested in the vector with some criterion of best direction regardless of its magnitude, which we may conveniently take as unity. Let us lump the C/ coefficients into the m x n matrix A and the n coefficients Vj into the vector x , hence 

Obviously, for m n, such a system has no exact solution, and, as before, the search is restricted to that for an estimate x of x. As the solution is only approximate, a residual error vector may be defined such that 

The least-square criterion suggests to minimize the modulus eTs of this error vector subject to the condition that the modulus iTi of the estimate is unity. The problem is therefore to minimize the sum c2 such as 

The solution to the problem is therefore the solution to the eigenvalue equation 

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