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

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## 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 s_{0} is contained in the null spaces of all s_{j}while the range of s_{0} 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 \( {Z_0} \)

## Keywords

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