Right divisors of numerator polynomial matrices and (A, B)-invariant subspaces
Antonis I. G. Vardulakis · International Journal of Control · 1980
Let [Ctilde](s)[Dtilde](s)−1 be a right coprime matrix fraction description of the (r × m) strictly proper transfer function matrix T(s) of a linear multivariable system Σ = (A, B, C). In this paper we establish a direct relation between (i) the algebraic concept of a right divisor C R(s) of the ‘numerator’ matrix [Ctilde](s), i.e. a (not necessarily square) polynomial matrix C L(s) that satisfies: [Ctilde](s) = C L(s)C R(s) for some polynomial matrix C L(s), and (ii) the geometric concept of an (A, B)-invariant subspace which is contained in the kernel of C. A special class of right divisors C R(s) of C(s) which correspond to (A, B)-invariant subspaces in ker C is characterized by a number of properties and a simple formula is presented which expresses the above correspondence. By generalizing the concept of a greatest right divisor (GRD) of a polynomial matrix to include also possibly non-square polynomial matrices, it is shown that GRDs of the numerator matrix C(s) correspond to the maximal (A, B)-invariant subspace in ker C.