The metric tensor and oblique projections

We now return to the very basic concept of vector length in vector spaces. In a Euclidian space spanned by a base of n orthogonal unit vectors eb the squared length l2 of a n-vector t is the quadratic form given by 

In a Euclidian space, the squared length of a vector is the sum of its squared 

We assume that we have shifted to a real new vector base, i.e., the vectors Sj are independent and B is non-singular. Now, we can go back to the v coordinates and write that, in a Euclidian space, vector length is an invariant 

For convenience, we will write w=B lv and call w( the ith coordinate of vector w in the new base of the vectors sj4 We also define the matrix G as 

The concept of metric tensor becomes central whenever distances and projections are considered, particularly when least-square criterion are used, a point that will be discussed in Chapter 5. Let us ask the frequently raised question of how to find an expression in terms of old coordinates (e.g., oxide proportions) for a projection made in the non-Euclidian space. This could be the case for finding oxide abundances of a basalt composition projected in the Yoder and Tilley tetrahedron, or the oxide abundance of a metamorphic rock composition projected into an ACF diagram assuming that quartz is present. 

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