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 $A\in {\mathcal{M}}_{m,n}(\mathbb{C})$ may be factored as the product of its conjugate transpose $A^{*}$ and a polynomial in either $A{A}^{*}$ 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.

摘要

本文提出了一种使用矩阵中的多项式来计算复杂矩阵的Moore-Penrose逆的方法。这种方法对所有矩阵均有效，并且不涉及频谱计算，这在矩阵大小较大时可能不可行。我们首先研究在何种条件下方阵的Moore-Penrose逆一个是多项式一个。作为应用，我们还看到任意矩阵的Moore-Penrose逆$\mathrm{一种}\in {\mathcal{中号}}_{米，ñ}（\mathbb{C}）$ 可能被认为是其共轭转置的乘积 ${\mathrm{一种}}^{*}$ 还有一个多项式 $\mathrm{一种}{\mathrm{一种}}^{*}$ 或者 ${\mathrm{一种}}^{*}\mathrm{一种}$。这篇文章是独立的，因此所有具有线性代数基础知识的读者都可以理解。我们已通过示例说明了大多数相关结果。