Invertible bases and root vectors for analytic matrix-valued functions

TL;DR

This paper explores invertible bases for submodules over certain rings, especially analytic functions, and applies this to define eigenvectors for analytic matrix functions even when they lack full rank.

Contribution

It introduces the concept of invertible bases for submodules over Bézout and elementary divisor domains, linking it to classical results and applying it to analytic matrix functions.

Findings

Submodules over Bézout domains have bases under certain conditions.

Invertible bases correspond to free pure submodules in elementary divisor domains.

Applied to analytic matrices, invertible bases enable defining eigenvectors without full rank.

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 B\'{e}zout 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 $\Omega \subseteq \mathbb{C}$ be either a connected compact set or a connected open set, and we specialize to $R=\mathcal{A}$, the ring of functions that are analytic on $\Omega$. We show that, for any matrix $A(z) \in \mathcal{A}^{m \times n}$, $\mathrm{ker} \ A(z) \cap \mathcal{A}^n$ is a free $\mathcal{A}$-module and admits an invertible basis, or equivalently a basis that is full rank upon evaluation at any $\lambda \in \Omega$. Finally, given $\lambda \in \Omega$, we use invertible bases to define and study maximal sets of root vectors at $\lambda$ for $A(z)$. This in particular allows us to define eigenvectors also for analytic matrices that do not have full column rank.