Operators and spaces associated to matrices with grades and their decompositions

We present results on decompositions of matrices with grades, i.e. matrices I with entries from a bounded ordered set L such as the real unit interval [0,1]. We consider decompositions of an n x m matrix I into a circular and triangular product A * B of an n x k matrix A and a k x m matrix B with k as small as possible. This problem generalizes the decomposition problem of Boolean factor analysis in which a decomposition of a binary matrix is sought into two binary matrices and which is a particular case of our setting when L has just two elements, namely 0 and 1. In our previous work, we proved that formal concepts of concept lattices associated to I are optimal factors for such decompositions. In this paper, we investigate concept-forming operators and concept lattices associated to decompositions of matrices and implications of these results.