Explicit solutions of the Yang–Baxter-like matrix equation for a diagonalizable matrix with spectrum contained in {1, α, 0}

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Abstract

Let A ∈ Cl × l be a diagonalizable matrix with spectrum contained in the set {1, α, 0}. In this paper we derive a general and explicit expression for the solutions X of the Yang–Baxter-like matrix equation AXA=XAX. The idea involves partitioning the matrices to discuss four block-matrix equations and utilizing the eigen-properties of the matrices obtained. When A is an idempotent matrix, the result generates the formula obtained in Mansour et al. (2017). We give examples to illustrate the validity of the results obtained in this note.

Introduction

Much research on solving the Yang–Baxter-like matrix equation (also called the star-triangle-like equation in statistical mechanics; see, e.g., in Part III of [8]),AXA=XAX,where A is a given square complex or real matrix and X is the unknown, has recently been seen in the literature (e.g., [1], [2], [3], [4], [5], [7], [9].) However, since the equation is nonlinear, it is generally not easy to find all the solutions for the equation when the matrix A is arbitrary and there have been only basic and partial answers to the problem and ‘very’ special cases were found to have been discussed so far. To the best of our knowledge, seeking for the solutions X satisfying AX=XA was discussed in [5] and the expressions of X were established for diagonalizable matrices A. And, in the class of diagonalizable matrices, the commuting and non-commuting solutions were given implicitly for idempotent matrices [1] and it is demonstrated that the solutions can be found by solving a system of three matrix equations of smaller size, but it seems little research has been conducted on how to obtain the explicit formulas of the solutions. In the case that A has rank one, the solutions have been obtained in [9]. Most recently, for a general idempotent matrix A, Mansour et al. [7] established an explicit formula for all the solutions of Eq. (1).

In this paper we derive an explicit formula for the solutions X of Eq. (1) for the diagonalizable matrix A with spectrum contained in the set {1, α, 0} (where α is any nonzero complex or real number), extending the research for which A is an idempotent matrix and A has rank one, respectively. In Section 2 we first derive some fundamental results for solving Eq. (1), where, based on the distinct eigenvalues of A, we partition the matrices to consider four block-matrix equations and then, by utilizing the algebraic properties of the matrices obtained, we establish the explicit expression for the solutions X. Examples are given to illustrate the configuration.

Throughout the note, the vectors/matrices are complex unless otherwise specified and V* stands for the transpose and conjugate of a vector/matrix V. Without confusion, for a nonsingular matrix Y, we sometimes write Y=(yi) and Y1=(yj), where yi and yj are the ith column vector of Y and the jth row vector of its inverse Y1, respectively. Except the notations addressed above, we will also use the following usual ones:

Section snippets

Finding the general solution for the equation

Before getting into the main discussion on solving the equation we need to establish some preliminaries.

Lemma 2.1

If Y ∈ Cm × m satisfies the equationY2+αY=(1α)2(Y4+αY3),α0(where α is any nonzero complex or real number), then its possible distinct eigenvalues are 0, α, 11α, 11α, and any Jordan block in the Jordan canonical form of Y has order at most 2 × 2. Specifically, (i) if 0 is an eigenvalue of Y then J(0) is diagonal; (ii) if α is an eigenvalue of Y then J(α) is diagonal if and only if (1α)α

Concluding remarks

In this note we derived an explicit expression for all solutions X of the Yang–Baxter-like matrix equation XAX=AXA, where A is a given diagonalizable matrix with distinct eigenvalues 1, α, 0. We considered the equation by partitioning the matrices according to the distinct eigenvalues of A. The idea behind the technique can be generalized to consider more general cases.

Finding the solutions to the Yang–Baxter-like matrix equation is a very challenging topic of research. we noticed that, given

Acknowledgments

The authors would like to thank Dr. Robert Acar for discussing some of the results obtained in the preliminary version of the paper. The authors are also indebted to the referees’ insightful comments and valuable suggestions towards improving the presentation of this paper. The research was supported partially by the Shenzhen University Research Foundation for Academic Leaders (SZU0000040242), The University of Puerto Rico at Mayaguez, and DIMACS (US NSF center at Rutgers, The State University

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