On the irreducibility of layered mixed matrices
Kazuo Murota · Linear and Multilinear Algebra · 1989
A matrix of the form is called a layered mixed matrix (or an LM-matrix) if the nonzero entries of Tare algebraically independent over the field to which the entries of Q belong. It is known that there exists a unique decomposition of an LM-matrix into irreducible blocks. The canonical block-triangular matrix with irreducible diagonal blocks is called the combinatorial canonical form (CCF) of the LM-matrix and plays a fundamental role in systems analysis. This paper gives a characterization of the irreducibility of an LM-matrix in terms of its determinant:A nonsingular LM-matrix is essentially irreducible iff its determinant is an irreducible polynomial in the nonzero entries of T. This is a substantial extension of a similar characterization of the full indecomposability of a formal incidence matrix due to H. J. Ryser.