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Abstract
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 R^{n} where R is a Bézout domain. Then, we define the concept of invertible basis of a submodule of R^{n}, 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 R^{n} . 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.
Original language  English 

Pages (fromto)  113 
Number of pages  13 
Journal  Electronic Journal of Linear Algebra 
Volume  40 
DOIs  
Publication status  Published  5 Jan 2024 
MoE publication type  A1 Journal articlerefereed 
Keywords
 Analytic function
 Eigenvector
 Free module
 Invertible basis
 Maximal set
 Minimal basis
 Pure submodule
 Root vector
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Noferini_Vanni_AoF_Project: Noferini Vanni Academy Project
Noferini, V., Quintana Ponce, M., Barbarino, G., Wood, R. & Nyman, L.
01/09/2020 → 31/08/2024
Project: Academy of Finland: Other research funding