RATIONAL SOLUTIONS OF THE MATRIX EQUATION p(X) = A

Gilbert J. Groenewald; Gerrit Goosen; Dawie Janse JANSE VAN RENSBURG; André C.M. Ran; Madelein Thiersen

doi:10.13001/ela.2024.8677

RATIONAL SOLUTIONS OF THE MATRIX EQUATION p(X) = A

Gilbert J. Groenewald, Gerrit Goosen^*, Dawie Janse JANSE VAN RENSBURG, André C.M. Ran, Madelein Thiersen

^*Corresponding author for this work

Mathematics

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

Abstract

We extend Theorem 1 of R. Reams, A Galois approach to m-th roots of matrices with rational entries, LAA, 258:187–194, 1997. Let p(λ) be any polynomial over ℚ, and let A ∈ M_n(ℚ) have irreducible characteristic polynomial f(λ) with degree n. We provide necessary and sufficient conditions for the existence of a solution X ∈ M_n(ℚ) of the polynomial matrix equation p(X) = A. Specifically, we find necessary and sufficient conditions for f(p(λ)) to have a factor of degree n over ℚ.

Original language	English
Pages (from-to)	564-573
Number of pages	10
Journal	Electronic Journal of Linear Algebra
Volume	40
Early online date	14 Aug 2024
DOIs	https://doi.org/10.13001/ela.2024.8677
Publication status	Published - Aug 2024

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© 2024, International Linear Algebra Society. All rights reserved.

Keywords

Matrix polynomial equation
Nonderogatory matrix
Rational solution

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title = "RATIONAL SOLUTIONS OF THE MATRIX EQUATION p(X) = A",

abstract = "We extend Theorem 1 of R. Reams, A Galois approach to m-th roots of matrices with rational entries, LAA, 258:187–194, 1997. Let p(λ) be any polynomial over ℚ, and let A ∈ Mn(ℚ) have irreducible characteristic polynomial f(λ) with degree n. We provide necessary and sufficient conditions for the existence of a solution X ∈ Mn(ℚ) of the polynomial matrix equation p(X) = A. Specifically, we find necessary and sufficient conditions for f(p(λ)) to have a factor of degree n over ℚ.",

keywords = "Matrix polynomial equation, Nonderogatory matrix, Rational solution",

author = "Groenewald, {Gilbert J.} and Gerrit Goosen and {JANSE VAN RENSBURG}, {Dawie Janse} and Ran, {Andr{\'e} C.M.} and Madelein Thiersen",

year = "2024",

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T1 - RATIONAL SOLUTIONS OF THE MATRIX EQUATION p(X) = A

AU - Groenewald, Gilbert J.

AU - Goosen, Gerrit

AU - JANSE VAN RENSBURG, Dawie Janse

AU - Ran, André C.M.

AU - Thiersen, Madelein

PY - 2024/8

Y1 - 2024/8

N2 - We extend Theorem 1 of R. Reams, A Galois approach to m-th roots of matrices with rational entries, LAA, 258:187–194, 1997. Let p(λ) be any polynomial over ℚ, and let A ∈ Mn(ℚ) have irreducible characteristic polynomial f(λ) with degree n. We provide necessary and sufficient conditions for the existence of a solution X ∈ Mn(ℚ) of the polynomial matrix equation p(X) = A. Specifically, we find necessary and sufficient conditions for f(p(λ)) to have a factor of degree n over ℚ.

AB - We extend Theorem 1 of R. Reams, A Galois approach to m-th roots of matrices with rational entries, LAA, 258:187–194, 1997. Let p(λ) be any polynomial over ℚ, and let A ∈ Mn(ℚ) have irreducible characteristic polynomial f(λ) with degree n. We provide necessary and sufficient conditions for the existence of a solution X ∈ Mn(ℚ) of the polynomial matrix equation p(X) = A. Specifically, we find necessary and sufficient conditions for f(p(λ)) to have a factor of degree n over ℚ.

KW - Matrix polynomial equation

KW - Nonderogatory matrix

KW - Rational solution

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EP - 573

JO - Electronic Journal of Linear Algebra

JF - Electronic Journal of Linear Algebra

ER -

RATIONAL SOLUTIONS OF THE MATRIX EQUATION p(X) = A

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