# On the construction of explicit solutions to the matrix equation X2AX = AXA

Aihua Li, Edward Mosteig

Research output: Contribution to journalArticle

### Abstract

In a previous article by Aihua Li and Duane Randall, the existence of solutions to certain matrix equations is demonstrated via nonconstructive methods. A recurring example appears in that work, namely the matrix equation AXA = X2AX, where A is a fixed, square matrix with real entries and X is an unknown square matrix. In this paper, the solution space is explicitly constructed for all 2×2 complex matrices using Gröbner basis techniques. When A is a 2×2 matrix, the equation AXA = X2AX is equivalent to a system of four polynomial equations. The solution space then is the variety defined by the polynomials involved. The ideal of the underlying polynomial ring generated by the defining polynomials plays an important role in solving the system. In our procedure for solving these equations, Gröbner bases are used to transform the polynomial system into a simpler one, which makes it possible to classify all the solutions. In addition to classifying all solutions for 2 × 2 matrices, certain explicit solutions are produced in arbitrary dimensions when A is nonsingular. In higher dimensions, Gröbner bases are extraordinarily computationally demanding, and so a different approach is taken. This technique can be applied to more general matrix equations, and the focus here is placed on solutions coming from a particular class of matrices.

Original language English 142-153 12 Electronic Journal of Linear Algebra 21 Published - 24 Nov 2010

### Fingerprint

Matrix Equation
Explicit Solution
Square matrix
Polynomial
Polynomial Systems
Polynomial equation
Polynomial ring
Higher Dimensions
Existence of Solutions
Classify
Transform
Unknown
Arbitrary

### Keywords

• Gröbner bases
• Ideal
• Matrix equation

### Cite this

@article{d1a6a59dc6b34c3ca771a219f5aec14d,
title = "On the construction of explicit solutions to the matrix equation X2AX = AXA",
abstract = "In a previous article by Aihua Li and Duane Randall, the existence of solutions to certain matrix equations is demonstrated via nonconstructive methods. A recurring example appears in that work, namely the matrix equation AXA = X2AX, where A is a fixed, square matrix with real entries and X is an unknown square matrix. In this paper, the solution space is explicitly constructed for all 2×2 complex matrices using Gr{\"o}bner basis techniques. When A is a 2×2 matrix, the equation AXA = X2AX is equivalent to a system of four polynomial equations. The solution space then is the variety defined by the polynomials involved. The ideal of the underlying polynomial ring generated by the defining polynomials plays an important role in solving the system. In our procedure for solving these equations, Gr{\"o}bner bases are used to transform the polynomial system into a simpler one, which makes it possible to classify all the solutions. In addition to classifying all solutions for 2 × 2 matrices, certain explicit solutions are produced in arbitrary dimensions when A is nonsingular. In higher dimensions, Gr{\"o}bner bases are extraordinarily computationally demanding, and so a different approach is taken. This technique can be applied to more general matrix equations, and the focus here is placed on solutions coming from a particular class of matrices.",
keywords = "Gr{\"o}bner bases, Ideal, Matrix equation",
author = "Aihua Li and Edward Mosteig",
year = "2010",
month = "11",
day = "24",
language = "English",
volume = "21",
pages = "142--153",
journal = "Electronic Journal of Linear Algebra",
issn = "1081-3810",
publisher = "International Linear Algebra Society",

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In: Electronic Journal of Linear Algebra, Vol. 21, 24.11.2010, p. 142-153.

Research output: Contribution to journalArticle

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T1 - On the construction of explicit solutions to the matrix equation X2AX = AXA

AU - Li, Aihua

AU - Mosteig, Edward

PY - 2010/11/24

Y1 - 2010/11/24

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AB - In a previous article by Aihua Li and Duane Randall, the existence of solutions to certain matrix equations is demonstrated via nonconstructive methods. A recurring example appears in that work, namely the matrix equation AXA = X2AX, where A is a fixed, square matrix with real entries and X is an unknown square matrix. In this paper, the solution space is explicitly constructed for all 2×2 complex matrices using Gröbner basis techniques. When A is a 2×2 matrix, the equation AXA = X2AX is equivalent to a system of four polynomial equations. The solution space then is the variety defined by the polynomials involved. The ideal of the underlying polynomial ring generated by the defining polynomials plays an important role in solving the system. In our procedure for solving these equations, Gröbner bases are used to transform the polynomial system into a simpler one, which makes it possible to classify all the solutions. In addition to classifying all solutions for 2 × 2 matrices, certain explicit solutions are produced in arbitrary dimensions when A is nonsingular. In higher dimensions, Gröbner bases are extraordinarily computationally demanding, and so a different approach is taken. This technique can be applied to more general matrix equations, and the focus here is placed on solutions coming from a particular class of matrices.

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