INVERTIBLE BASES AND ROOT VECTORS FOR ANALYTIC MATRIX-VALUED FUNCTIONS

We revisit the concept of a minimal basis through the lens of the theory of modules over a commutative ring R. We first review the conditions for the existence of a basis for submodules of Rⁿ where R is a Bézout domain. Then, we define the concept of invertible basis of a submodule of Rⁿ, and when R is an elementary divisor domain, we link it to the Main Theorem of G. D. Forney Jr. [SIAM J. Control, 13:493–520, 1975]. Over an elementary divisor domain, the submodules admitting an invertible basis are precisely the free pure submodules of Rⁿ . As an application, we let Ω ⊆ C be either a connected compact set or a connected open set, and we specialize to R = A(Ω), the ring of functions that are analytic on Ω. We show that, for any matrix A(z) ∈ A(Ω)^m×n, ker A(z) ∩ A(Ω)ⁿ is a free A(Ω)-module and admits an invertible basis, or equivalently a basis that is full rank upon evaluation at any λ ∈ Ω. Finally, given λ ∈ Ω, we use invertible bases to define and study maximal sets of root vectors at λ for A(z). This in particular allows us to define eigenvectors also for analytic matrices that do not have full column rank.