An elementary description of partial indices of rational matrix functions

Symbolic Computation Applied to the Study of the Kernel of Special Classes of Paired Singular Integral Operators

“…Any non-singular rational matrix function f ∈ [R( )] n,n admits a left (right) factorization of the form (5), where…”

“…For the case when ϕ ∈ H r,θ , we designed the generalized factorization algorithm [AFact] [11,17] that computes a left generalized factorization of factorable essentially bounded second-order matrix functions of type (18), for any general inner function θ. In particular, the [AFact] algorithm allows us to know if a matrix function of the class (18) admits, or not, a left generalized factorization (5). Moreover, if A γ (ϕ) is factorable, the algorithm allows us to determine if the generalized factorization is canonical or non-canonical, and it gives us an explicit left generalized factorization of the matrix function.…”

“…Both algorithms were implemented using the computer algebra system Mathematica. On the determination of the partial indices, some developments have also been made but, even in the rational case (and in recent publications), the methods are difficult to apply and were not designed to be implemented on a computer (see, for instance, [5,8,35,49]). In addition, the vast majority of explicit analytical factorization methods depends on the knowledge of the zeros of scalar functions.…”