On the Concept of Invertibility for Sequences of Complex $p \times q$-matrices and its Application to Holomorphic $p \times q$-matrix-valued Functions

Abstract

The main topic of this paper is the invertibility of finite and infinite sequences of complex$p \times q$-matrices.This concept was previously considered in the mathematical literature for the special case in which p=q, under certain regularity conditions, in the context of matricial power series inversion. The problem of describing all (finite and infinite) invertible sequences of complex$p \times q$-matrices leads directly to the class of “first term dominant” sequences $(s_j)^{k}_{j=0}$of complex $p \times q$-matrices.These sequences have the property that the null space of s₀ is contained in the null spaces of all s_jwhile the range of s₀ encompasses the range of every s_j The inverse sequence $(s^{\ddag}_j)^{k}_{j=0}$ in $\mathbb{C}^{q\times p}$ for an invertible sequence $(s_j)^{k}_{j=0}$in $\mathbb{C}^{q\times p}$ is then constructed. This leads, in conjunction with the concept of power series inversion, to a generalrecursive method for constructing a reciprocal sequence $(s^{\sharp}_j)^{k}_{j=0}$ in $\mathbb{C}^{q\times p}$ to any given sequence $(s_j)^{k}_{j=0}$ in $\mathbb{C}^{q\times p}$ It is shown that if $(s_j)^{k}_{j=0}$ is invertible, then its inverse and reciprocal sequences coincide, i.e.,$(s^{\ddag}_j)^{k}_{j=0}$ = $(s^{\sharp}_j)^{k}_{j=0}$.Using reciprocal sequences allows for interesting new approaches to a number of fascinating problems in matricial complex analysis. This paper considers the holomorphicity of the Moore-Penrose inverse of a $p \times q$ -matrixfunction, using an approach based on analyzing the structure of Taylor coefficient sequences. A main result of this paper states that the Moore-Penrose inverse $F^{\dag}$ of a complex $p \times q$-matrix-function F which is holomorphic in an open disk K of the complex plane $\mathbb{C}$ is holomorphic in K if, and only if, the Taylor-McLaurin coefficient sequence $(s_j)^{k}_{j=0}$ for F at the center ${Z_0}$ of K is invertible. When this is the case, the reciprocal sequence $(s^{\ddag}_j)^{k}_{j=0}$ to $(s_j)^{k}_{j=0}$ is the Taylor-McLaurin coefficient sequence for $F^{\dag}$ in ${Z_0}$

Mathematics Subject Classification (2010).

30E05;

15A09.