Solutions of the matrix equation p(X)=A, with polynomial function p(λ) over field extensions of Q

G. J. Groenewald; D. B. Janse van Rensburg; A. C.M. Ran; F. Theron; M. van Straaten

doi:10.1016/j.laa.2023.01.028

Solutions of the matrix equation p(X)=A, with polynomial function p(λ) over field extensions of Q

G. J. Groenewald, D. B. Janse van Rensburg, A. C.M. Ran^*, F. Theron, M. van Straaten

^*Corresponding author for this work

Mathematics

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

Abstract

Let H be a field with Q⊆H⊆C, and let p(λ) be a polynomial in H[λ], and let A∈H^n×n be nonderogatory. In this paper we consider the problem of finding a solution X∈H^n×n to p(X)=A. A necessary condition for this to be possible is already known from a paper by M.P. Drazin: Exact rational solutions of the matrix equation A=p(X) by linearization. Under an additional condition we provide an explicit construction of such solutions. The similarities and differences with the derogatory case will be discussed as well. One of the tools needed in the paper is a new canonical form, which may be of independent interest. It combines elements of the rational canonical form with elements of the Jordan canonical form.

Original language	English
Pages (from-to)	107-138
Number of pages	32
Journal	Linear Algebra and its Applications
Volume	665
Early online date	3 Feb 2023
DOIs	https://doi.org/10.1016/j.laa.2023.01.028
Publication status	Published - 15 May 2023

Bibliographical note

Funding Information:
This work is based on research supported in part by the DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) and in part by the National Research Foundation of South Africa (Grant Numbers 145688 and 2022-012-ALG-ILAS ). Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the CoE-MaSS.

Publisher Copyright:
© 2023 The Author(s)

Funding

This work is based on research supported in part by the DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) and in part by the National Research Foundation of South Africa (Grant Numbers 145688 and 2022-012-ALG-ILAS ). Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the CoE-MaSS.

Keywords

Canonical forms
Companion matrices
Linear matrix equations
Matrices over field extensions of the rationals
Nonderogatory matrices
Solutions of polynomial matrix equations

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abstract = "Let H be a field with Q⊆H⊆C, and let p(λ) be a polynomial in H[λ], and let A∈Hn×n be nonderogatory. In this paper we consider the problem of finding a solution X∈Hn×n to p(X)=A. A necessary condition for this to be possible is already known from a paper by M.P. Drazin: Exact rational solutions of the matrix equation A=p(X) by linearization. Under an additional condition we provide an explicit construction of such solutions. The similarities and differences with the derogatory case will be discussed as well. One of the tools needed in the paper is a new canonical form, which may be of independent interest. It combines elements of the rational canonical form with elements of the Jordan canonical form.",

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AU - Ran, A. C.M.

AU - Theron, F.

AU - van Straaten, M.

N1 - Funding Information: This work is based on research supported in part by the DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) and in part by the National Research Foundation of South Africa (Grant Numbers 145688 and 2022-012-ALG-ILAS ). Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the CoE-MaSS. Publisher Copyright: © 2023 The Author(s)

PY - 2023/5/15

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N2 - Let H be a field with Q⊆H⊆C, and let p(λ) be a polynomial in H[λ], and let A∈Hn×n be nonderogatory. In this paper we consider the problem of finding a solution X∈Hn×n to p(X)=A. A necessary condition for this to be possible is already known from a paper by M.P. Drazin: Exact rational solutions of the matrix equation A=p(X) by linearization. Under an additional condition we provide an explicit construction of such solutions. The similarities and differences with the derogatory case will be discussed as well. One of the tools needed in the paper is a new canonical form, which may be of independent interest. It combines elements of the rational canonical form with elements of the Jordan canonical form.

AB - Let H be a field with Q⊆H⊆C, and let p(λ) be a polynomial in H[λ], and let A∈Hn×n be nonderogatory. In this paper we consider the problem of finding a solution X∈Hn×n to p(X)=A. A necessary condition for this to be possible is already known from a paper by M.P. Drazin: Exact rational solutions of the matrix equation A=p(X) by linearization. Under an additional condition we provide an explicit construction of such solutions. The similarities and differences with the derogatory case will be discussed as well. One of the tools needed in the paper is a new canonical form, which may be of independent interest. It combines elements of the rational canonical form with elements of the Jordan canonical form.

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Solutions of the matrix equation p(X)=A, with polynomial function p(λ) over field extensions of Q

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