A New Generalization of the Liouville-Jacobi Identity

TL;DR

This paper proves a generalized version of the Liouville-Jacobi Identity for determinants of solutions to a specific class of linear matrix differential equations, extending classical identities to nonhomogeneous cases.

Contribution

It introduces a new generalization of the Liouville-Jacobi Identity applicable to nonhomogeneous linear matrix differential equations with variable coefficients.

Findings

Established a generalized identity for determinants of matrix solutions.

Extended classical Liouville-Jacobi results to nonhomogeneous equations.

Provides a theoretical foundation for analyzing matrix differential equations.

Abstract

A generalized Liouville-Jacobi Identity is proved for the determinant $\det{X(t)}$ of a solution $X(t)$ to the linear nonhomogeneous first-order matrix differential equation with left- and right-coefficient matrices $\ \frac{{\rm d}}{{\rm d}t} X(t) + A(t)X(t) + X(t)B(t)= F(t), \ X(t_0)=X_0.$