Computing Moore–Penrose inverses with polynomials in matrices

Abstract

This article proposes a method for computing the Moore–Penrose inverse of a complex matrix using polynomials in matrices. Such a method is valid for all matrices and does not involve spectral calculation, which could be infeasible when the size of the matrix is large. We first study under which conditions the Moore–Penrose inverse of a square matrix A is a polynomial in A. As an application, we also see that the Moore–Penrose inverse of an arbitrary matrix A∈Mm,n(C) may be factored as the product of its conjugate transpose A∗ and a polynomial in either AA∗ or A∗A. The article is self-contained so that it can be understood by all readers with a basic knowledge of linear algebra. We have illustrated most of the relevant results with examples.