Polar decompositions of quaternion matrices in indefinite inner product spaces

Gilbert J. Groenewald; Dawie B.Janse VAN RENSBURG; André C.M. Ran; Frieda Theron; Madelein VAN STRAATEN

doi:10.13001/ela.2021.6411

Polar decompositions of quaternion matrices in indefinite inner product spaces

Gilbert J. Groenewald, Dawie B.Janse VAN RENSBURG, André C.M. Ran, Frieda Theron, Madelein VAN STRAATEN^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to Journal › Article › Academic › peer-review

Abstract

Polar decompositions of quaternion matrices with respect to a given indefinite inner product are studied. Necessary and sufficient conditions for the existence of an H-polar decomposition are found. In the process, an equivalent to Witt’s theorem on extending H-isometries to H-unitary matrices is given for quaternion matrices.

Original language	English
Pages (from-to)	659-670
Number of pages	12
Journal	Electronic Journal of Linear Algebra
Volume	37
Early online date	29 Oct 2021
DOIs	https://doi.org/10.13001/ela.2021.6411
Publication status	Published - 2021

Bibliographical note

Funding Information:
†Department of Mathematics and Applied Mathematics, Research Focus: Pure and Applied Analytics, North-West University, Private Bag X6001, Potchefstroom 2520, South Africa ([email protected], [email protected], [email protected], [email protected]). Supported by a grant from DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS).

Publisher Copyright:
© 2021, International Linear Algebra Society. All rights reserved.

Keywords

Extending isometries
H-polar decompositions
Indefinite inner product
Quaternion matrices
Square roots of matrices
Witt’s theorem

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10.13001/ela.2021.6411

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title = "Polar decompositions of quaternion matrices in indefinite inner product spaces",

abstract = "Polar decompositions of quaternion matrices with respect to a given indefinite inner product are studied. Necessary and sufficient conditions for the existence of an H-polar decomposition are found. In the process, an equivalent to Witt{\textquoteright}s theorem on extending H-isometries to H-unitary matrices is given for quaternion matrices.",

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note = "Funding Information: †Department of Mathematics and Applied Mathematics, Research Focus: Pure and Applied Analytics, North-West University, Private Bag X6001, Potchefstroom 2520, South Africa ([email protected], [email protected], [email protected], [email protected]). Supported by a grant from DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS). Publisher Copyright: {\textcopyright} 2021, International Linear Algebra Society. All rights reserved.",

year = "2021",

doi = "10.13001/ela.2021.6411",

language = "English",

volume = "37",

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AU - Groenewald, Gilbert J.

AU - VAN RENSBURG, Dawie B.Janse

AU - Ran, André C.M.

AU - Theron, Frieda

AU - VAN STRAATEN, Madelein

N1 - Funding Information: †Department of Mathematics and Applied Mathematics, Research Focus: Pure and Applied Analytics, North-West University, Private Bag X6001, Potchefstroom 2520, South Africa ([email protected], [email protected], [email protected], [email protected]). Supported by a grant from DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS). Publisher Copyright: © 2021, International Linear Algebra Society. All rights reserved.

PY - 2021

Y1 - 2021

N2 - Polar decompositions of quaternion matrices with respect to a given indefinite inner product are studied. Necessary and sufficient conditions for the existence of an H-polar decomposition are found. In the process, an equivalent to Witt’s theorem on extending H-isometries to H-unitary matrices is given for quaternion matrices.

AB - Polar decompositions of quaternion matrices with respect to a given indefinite inner product are studied. Necessary and sufficient conditions for the existence of an H-polar decomposition are found. In the process, an equivalent to Witt’s theorem on extending H-isometries to H-unitary matrices is given for quaternion matrices.

KW - Extending isometries

KW - H-polar decompositions

KW - Indefinite inner product

KW - Quaternion matrices

KW - Square roots of matrices

KW - Witt’s theorem

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JF - Electronic Journal of Linear Algebra

ER -

Polar decompositions of quaternion matrices in indefinite inner product spaces

Abstract

Bibliographical note

Keywords

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