<abstract xmlns="http://www.w3.org/1999/xhtml">Matrix expressions composed by generalized inverses can generally be written as <italic>f</italic>(<italic>A</italic>−1, <italic>A</italic>−2, . . ., <italic>A</italic>−<italic>k</italic>), where <italic>A</italic>1, <italic>A</italic>2, . . ., <italic>A</italic><italic>k</italic> are a family of given matrices of appropriate sizes, and (·)− denotes a generalized inverse of matrix. Once such an expression is given, people are primarily interested in its uniqueness (invariance property) with respect to the choice of the generalized inverses. As such an example, this article describes a general method for deriving necessary and sufficient conditions for the matrix equality <italic>A</italic>1<italic>A</italic>−2<italic>A</italic>3<italic>A</italic>−4<italic>A</italic>5 = <italic>A</italic> to always hold for all generalized inverses <italic>A</italic>−2 and <italic>A</italic>−4 of <italic>A</italic>2 and <italic>A</italic>4 through use of the block matrix representation method and the matrix rank method, and discusses some special cases of the equality for different choices of the five matrices.</abstract>

Matrix expressions composed by generalized inverses can generally be written as f(A−1, A−2, . . ., A−k), where A1, A2, . . ., Ak are a family of given matrices of appropriate sizes, and (·)− denotes a generalized inverse of matrix. Once such an expression is given, people are primarily interested in its uniqueness (invariance property) with respect to the choice of the generalized inverses. As such an example, this article describes a general method for deriving necessary and sufficient conditions for the matrix equality A1A−2A3A−4A5 = A to always hold for all generalized inverses A−2 and A−4 of A2 and A4 through use of the block matrix representation method and the matrix rank method, and discusses some special cases of the equality for different choices of the five matrices.

Invariance property of a five matrix product involving two generalized inverses

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

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{"article-title":"Invariance property of a five matrix product involving two generalized inverses"}

Matrix expressions composed by generalized inverses can generally be written as f(A−1, A−2, . . ., A−k), where A1, A2, . . ., Ak are a family of given matrices...

Invariance property of a five matrix product involving two generalized inverses

Published Online: Apr 13, 2021

Page range: 83 - 92

Received: Nov 01, 2019

Accepted: May 04, 2020

DOI: https://doi.org/10.2478/auom-2021-0006

Keywords
Generalized inverse, matrix product, matrix equation, rank formula, invariance

© 2021 Bo Jiang et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Invariance property of a five matrix product involving two generalized inverses

Published Online: Apr 13, 2021

Page range: 83 - 92

Received: Nov 01, 2019

Accepted: May 04, 2020

DOI: https://doi.org/10.2478/auom-2021-0006

KeywordsGeneralized inverse, matrix product, matrix equation, rank formula, invariance

© 2021 Bo Jiang et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Generalized inverse, matrix product, matrix equation, rank formula, invariance