Modeling and Analysis of Compositional Data

4.9 Matrix operations in the simplex

Many operations in real spaces are easier expressed in matrix notation. As the simplex is an Euclidean space, matrix notations may also be useful for compositions. However, there is a fundamental difference between the real space and the simplex: in the first case, one can identify an array of scalar values with a vector of the space. But in the case of the simplex, a vector of real constants may not be identifiable with a composition. This produces two kinds of matrix products that are introduced in this section. The first is simply the expression of a perturbation-linear combination of compositions, which appears as a power multiplication of a real vector by a compositional matrix in which rows are compositions. The second one is the expression of a linear transformation in the simplex: a composition in coordinates is transformed by a matrix product, and the result is expressed back as a composition. Each component of the result can be shown to be equivalent to a product of powered components of the original composition, where these powers can be arranged in a square matrix that identifies the transformation. The power matrix implied in this case is not unique, a consequence of the nature of compositions as equivalence classes.

4.9.1 Perturbation-linear combination of compositions

For a row vector of scalars and an array of row compositions ...