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

  • Bernd FritzscheEmail author
  • Bernd Kirstein
  • Conrad Mädler
  • Tilo Schwarz
Chapter
  • 448 Downloads
Part of the Operator Theory: Advances and Applications book series (OT, volume 226)

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 sjwhile the range of s0 encompasses the range of every sj 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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Copyright information

© Springer Basel 2012

Authors and Affiliations

  • Bernd Fritzsche
    • 1
    Email author
  • Bernd Kirstein
    • 1
  • Conrad Mädler
    • 1
  • Tilo Schwarz
    • 1
  1. 1.Mathematisches InstitutUniversität LeipzigLeipzigGermany

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