On the area of feasible solutions for rank-deficient problems: II. The geometric construction

A multivariate curve resolution problem is said to suffer from a rank-deficiency if the rank of the spectral data matrix is less than the number of the involved chemical species. A rank-deficiency is caused by linearly dependent (in the sense of linear algebra) concentration profiles or spectra of the pure components. The rank-loss is propagated to the spectral mixture data according to the bilinear Lambert–Beer superposition. This work deals with factor ambiguities for rank-deficient problems and presents an approach for the geometric construction of the area of feasible solutions (AFS). The focus is on the case that the rank-deficient matrix factor has the rank three and the number of chemical species equals at least four. The AFS construction works with polygons tightly enclosing the inner polygon, namely with quadrangles in the case of four chemical species, pentagons for five species and so on. •Rank-deficiency complicates the MCR analysis of factor ambiguities.•Such ambiguities are analyzed from a geometric point of view.•The Borgen-Kowalski triangle rotation is extended to polygon rotations.