Horizontal Principal Structure of Layered Mixed Matrices: Decomposition of Discrete Systems by Design-Variable Selections

A matrix $A=\bigl(\begin{smallmatrix}Q\\T\end{smallmatrix}\bigr)$ is called a layered mixed matrix (LM-matrix) if the set of nonzero entries of $T$ is algebraically independent over the field to which the entries of $Q$ belong. This concept has been proposed as a mathematical tool for describing discrete physical/engineering systems. It is known that there uniquely exists a finest block-triangularization of an LM-matrix, which is called the combinatorial canonical form (CCF). In this paper, associated with an LM-matrix we introduce a new submodular function $q$ characterizing its rank. This submodular function $q$ is defined on a modular lattice. It will be shown that the principal structure of $q$ gives the coarsest decomposition of the row side that is finer than any decomposition induced by the CCF of the submatrix consisting of a base of the column vectors of $A$. This gives a best possible bound on the extent to which the whole system can be decomposed by a suitable choice of design variables.