Linear constraints the closure condition

Since many geochemical units are concentrations of fractions which sum up to unity, let us first demonstrate a useful statement. A vector is normalized when its components 

In most cases of interest, however, the system represented by equation (5.3.3) is overdetermined and we must enforce the closure condition with a different method. Let us return to a standard mass-balance least-square problem, such as, for instance, calculating the mineral abundances from the whole-rock and mineral chemical compositions. If xu x2,.. -,x are the mineral fractions, which may be lumped together in a vector x, the closure condition 

Arranging the whole-rock mineral concentrations for each element i (i=l.m) in a vector y, and putting the concentration of element i in the phase j at the ith row and jth column of the matrix Amxn (mineral matrix), the usual overdetermined system is obtained 

Since each term in c2 is a scalar and therefore symmetric, equation (5.3.7) may be rewritten as 

The second and third scalar terms of c2 are the transpose of each other and are therefore equal, hence 

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