On the Yang-Baxter-like matrix equation for rank-two matrices

1 Introduction

Recently [1], the authors have found all the solutions of the Yang-Baxter-like matrix equation

where the given n × n complex matrix A = PQ^T, with two n × 2 matrices P and Q, satisfies the assumption that the 2 × 2 matrix Q^TP is nonsingular. In the above situation, the eigenvalue 0 of A is semi-simple with multiplicity n − 2. This leaves the case unsolved that Q^TP is singular, which makes it much more challenging to solve the corresponding matrix equation. The purpose of this paper is to find all the solutions of (1) for the remaining case that the algebraic multiplicity of the eigenvalue 0 of A is more than n − 2.

The equation (1) has a similar format to the classical Yang-Baxter equation [2]. Yang [3] in 1967 first considered a one dimensional quantum mechanical many body problem with a combination of delta functions as the potential and found a factorization of the scattering matrix, and the Yang-Baxter equation was obtained as a consistence property for the factorization. Then Baxter in 1972 solved an eight-vertex model in statistical mechanics, resulting in a similar matrix equation, which, together with that from [3], was first called the Yang-Baxter equation by Russian researchers at the end of the 1970s. Since then the Yang-Baxter equation has been extensively investigated by mathematicians and physicists in knot theory, braid group theory, and quantum group theory in the past thirty years; see, e.g., [2, 4-8] and the references therein. In the past several years, the quadratic matrix equation (1) has been studied with linear algebra techniques; see, for example, the references [9-14].

Solving (1) is a tough job in general since we need to solve a large system of n² quadratic equations with n² variables if we multiply out its both sides. By restricting the task to only finding the solutions that commute with A, several solution results have been obtained in [10] for matrices A of special Jordan forms, and a more general result was proved in [15] for the class of diagonalizable matrices. However, no general result has been found so far for non-commuting solutions for arbitrary matrices. Thus, it is our hope to find all the solutions of (1) for general matrices A. When A is a rank one matrix, all the solutions of (1) have been found in [13]. The case of A being of rank-two turns out to be much more tedious to analyze, and some special cases have been solved in our previous paper [1]. In the current paper we continue to study the rank two case to find all the solutions for the remaining Jordan form structure of A.

Matrices A of rank at most two in the equation (1) have appeared in the classic Yang-Baxter equation (see, e.g., the references in [2, 5]). For example, the two classes of 4 × 4 matrices

with certain values of the given parameters have been studied in [16-18] for completely integrable systems and inverse scattering problems. Each matrix in the second class is actually the tensor product T ⊗ I₂ of the 2 × 2 matrix T = [t_ij] with the 2 × 2 identity matrix I₂, which constitutes a basic operation in the context of the classical Yang-Baxter equation. Our complete solutions to the quadratic matrix equation with a given rank-two matrix are expected to find applications in such physical applications. We shall pick up one matrix from (2) to apply our result in Section 5.

It is well known [10] that solving the Yang-Baxter-like matrix equation for any given matrix A is equivalent to solving the same equation with A replaced by a matrix that is similar to A, and all the solutions of the first equation are similar to those of the second one with the same similarity matrix. Thus, solving (1) for the given A can be reduced to solving the same equation with the Jordan form of A. Our approach in this paper will follow the above principle. That is, since any matrix is similar to its simplest possible canonical form J, the Jordan form of the matrix, which is called a Jordan matrix here for simplicity, is to be found. For the purpose of the present paper that will supplement the work of [1], we shall solve the following simpler Yang-Baxter-like matrix equation

with J satisfying the conditions:

1. J is a Jordan matrix of rank-2.

2. The eigenvalue 0 of J has algebraic multiplicity at least n − 1.

If we can find the solutions of (3) for such Jordan matrices, then the solutions of (1) are available immediately for any A that is similar to J.

It turns out that the Jordan matrices J that satisfy the conditions (i) and (ii) above are

such that Λ is one of the following three matrices

and the diagonal block 0 in (4) is either (n − 3) × (n − 3) or (n − 4) × (n − 4) accordingly. With respect to each of the three cases for J, there exists a corresponding nonsingular similarity matrix W = [w₁, …, w_n] connecting A to J, so that

More specifically, in the case that Λ = Λ₁, the columns w₁,…, w_{n − 2} of W are eigenvectors of A and the columns w_{n − 1} and w_n of W are generalized eigenvectors of A with degrees 2 and 3 respectively, namely

all associated with eigenvalue 0. When Λ = Λ₂, the first n − 2 columns of W are eigenvectors of A associated with eigenvalue 0, the column w_{n − 1} is a generalized eigenvector of degree 2 with respect to eigenvalue 0, and the column w_n is an eigenvector of A associated with eigenvalue λ ≠ 0. If Λ = Λ₃, then the columns w₁, …, w_{n − 3}, and w_{n − 1} of W are eigenvectors of A, and the columns w_{n − 2} and w_n of W are generalized eigenvectors of degrees 2, all with respect to eigenvalue 0.

In the next three sections we look for the solutions of the Yang-Baxter-like matrix equation (1) when the corresponding Jordan matrix J of A is given by (4) with Λ = Λ₁, Λ₂, Λ₃, respectively; such cases will be referred to as type I, type II, and type III for the given matrix A. We present two examples of our solution result in Section 5 and conclude with Section 6.

2 Solutions when A is of type I

In this and subsequent sections we let A = PQ^T in (1) with P = [p₁, p₂] and Q = [q₁, q₂] of rank-2 such that det Q^TP = 0. Let J be the Jordan form of A given by (4), where the diagonal zero sub-matrix 0 is either (n − 3) × (n − 3) and the sub-matrix Λ = Λ₁ or Λ₂ defined by (5), or the zero sub-matrix 0 is (n − 4) × (n − 4) and Λ = Λ₃ in (5). As pointed out in Section 1, it is enough to find all the solutions of the equation (3) with J the Jordan form of A, so we just focus on solving (3).

Let Y be partitioned the same way as J into the 2 × 2 block matrix

where M is (n − 3) × (n − 3) or (n − 4) × (n − 4), U = [u₁, u₂, u₃] or [u₁, u₂, u₃, u₄], V = [v₁, v₂, v₃] or [v₁, v₂, v₃, v₄], and K is 3 × 3 or 4 × 4, depending on the size of Λ. Then (3) with J partitioned by (4) is

which is equivalent to the system

In the current section we assume that J = diag (0, Λ₁), and the other two cases that J = diag (0, Λ₂) and J = diag (0, Λ₃) will be investigated in Sections 3 and 4, respectively. So we solve (8) with Λ = Λ₁. Because of the special zero structure of Λ₁, the two unknown vectors u₃ and v₁ actually do not appear at all in the above system, so they always appear as free variables in the solutions. In addition, the first equation is independent of K and is in fact

The last equation of (8),

is itself a Yang-Baxter-like matrix equation of small size when Λ = Λ_j with j = 1, 2, 3, and finding all of its solutions is the first step for solving (8).

3 Solutions when A is of type II

We now consider the second case that the Jordan form of the matrix A is J = diag (0, Λ₂), so we solve (8) with Λ = Λ₂. Clearly, the structure of Λ makes u₂ and v₁ free vectors in all the solutions of (8), and the first equation of (8) is now

4 Solutions when A is of type III

Unlike the previous two cases that involve only 3 × 3 matrices Λ₁ and Λ₂, the third case that we shall study involves the 4 × 4 matrix Λ₃. The zero structure of Λ₃ ensures that u₂, u₄, v₁,v₃ are not present in the equations, so they are free vectors in the solutions. The following lemma gives all the solutions of the last equation of (8).