A Note on Minimal Polynomial Operators for Singular Matrix Systems

While the concept and use of minimal polynomials is classical in the context of linear algebra, their direct application to derive a matrix operator for singular matrices are not widely utilized.In this paper, we develop a residual matrix derived directly from the coefficients of the minimal polynomial of a singular matrix.We introduce the necessary properties of the proposed residual matrix in order to solve singular systems.In particular, we establish the necessary and sufficient conditions in order to solve singular matrix equations and initial value problems involving singular coefficient matrices.This approach forms an algebraic alternative to traditional methods that rely on pseudoinverses or spectral decompositions with potential advantages in both simplicity and computational efficiency.