Factorization of a Matrix Differential Operator Using Functions in its Kernel

TL;DR

This paper generalizes the factorization of scalar differential operators to matrix coefficient differential operators, including cases with singular leading coefficients, using functions in its kernel.

Contribution

It introduces a straightforward method to factor matrix differential operators with singular leading coefficients based on kernel functions, extending classical scalar results.

Findings

Provides a generalization of scalar operator factorization to matrix operators

Applicable even when the leading coefficient is singular

Offers a practical approach for operator decomposition

Abstract

Just as knowing some roots of a polynomial allows one to factor it, a well-known result provides a factorization of any scalar differential operator given a set of linearly independent functions in its kernel. This note provides a straight-forward generalization to the case of matrix coefficient differential operators that applies even in the case that the leading coefficient is singular.