1. Introduction
Nonlinear matrix equations arise in many scientific and engineering fields. Seeking their solutions is a difficult task [
1]. In this paper, we are interested in solving the nonlinear matrix equation of the form
where
A is a given nonzero complex matrix. Matrix Equation (
1) is called the Yang-Baxter-like matrix equation (see [
2,
3,
4,
5,
6,
7,
8,
9]), termed YBME for short, since it originates from physics. Matrix Equation (
1) has the same form as the well-known Yang–Baxter equation, which first arose in two independent papers by Yang [
10] and Baxter [
11]. Recently, Kumar et al. [
12] investigated new solution sets for the Yang-Baxter-like matrix equation by using a class of generalized outer inverses of a matrix. Jiang et al. [
13] proposed a zeroing neural network dynamical system approach for solving the time-varying Yang–Baxter matrix equation.
It is evident that YBME (
1) is a symmetric matrix equation concerning a known matrix
A and an unknown matrix
X. Although YBME (
1) looks simple in format, it is not easy to solve for a general matrix
A since it is equivalent to solving a quadratic system of
equations in
variables. So far, only for a few matrices with a special structure (f.g. [
2,
3,
4,
5,
6]) can all commutable solutions of Equation (
1) be obtained, i.e., solutions satisfying
.
In this paper, our aim is to find all solutions of YBME (
1) with an idempotent matrix
A, that has already been provided in [
5]. However, the idea of our approach is novel. We do not need to use and compute the Jordan canonical form of
A as in some previous works, such as [
2,
3,
4,
5,
6,
7,
8,
9]. Our analysis and method to construct the solution set are based on simple sufficient and necessary conditions that we derive for a matrix to be a nontrivial solution of YBME (
1). In particular, we construct all commuting solutions.
The rest of the paper is organized as follows. In
Section 2, we present some properties of the idempotent matrices. In
Section 3, we first derive simple sufficient and necessary conditions for a matrix being a nontrivial (commuting, non-commuting) solution of YBME (
1), and then analyze and construct the set that contains all the (commuting, non-commuting) solutions of YBME (
1). Finally, in
Section 4, we provide an example.
2. Preliminaries
In this section, we first provide some properties of the idempotent matrices that are needed for our analysis later.
Let
A be an
idempotent matrix with rank
r, satisfying
and
. We assume that
to avoid the two trivial cases
or
. In fact, when
, YBME (
1) is zero. When
, YBME (
1) reduces to
, whose solution is any idempotent matrix. That is to say, if we denote the set of all the solutions of the matrix for equation
as
we know that
is the set of all the idempotent matrices of order
n.
It is widely known that any square matrix
A with rank
r has the spectral decomposition in the form
where
and
are of full rank. For notational convenience, in the following, we will denote
where
and
form the basis of the null space of
and
, respectively. Therefore, we have
and
.
Lemma 1. Let , where U and V are two () complex matrices of full column rank; then, A is a singular idempotent matrix if and only if .
Proof. Since , . Thus, A is an idempotent matrix if and only if . □
According to Lemma 1, we easily verify that
where
Now, we begin to find solutions of YBME (
1) with an idempotent matrix.
Lemma 2. Let be an idempotent matrix with rank r, and let E be a matrix, then is a (commuting) solution of YBME (1) if and only if E is an ( ) idempotent matrix. Proof. Let
, notice that
. Thus we have
So,
X is a solution of YBME (
1) if and only if
, i.e.,
E is an idempotent matrix. Further, since
,
X commutes with
A. □
Lemma 3. Let be an idempotent matrix with rank r, then for any matrix F, is a commuting solution of YBME (1). Proof. In fact, let
, notice that
and
, so we have
So,
X is a commuting solution of YBME (
1). □
3. All the Solutions
In this section, we find all the solutions
X of YBME (
1) with an idempotent matrix based on simple sufficient and the necessary conditions for a matrix being a nontrivial (commuting, non-commuting) solution. We first provide a lemma.
Lemma 4. If where U and V are two complex matrices of full rank such that , then the two matricesare nonsingular. Proof. To show the matrix is nonsingular, we only need to show that the linear system only has a zero solution. In fact, since and , we have . The linear system becomes . Further, since is of full column rank, we derive from that . So, has only a zero solution; thus, P is nonsingular. Similarly, we can show that Q is nonsingular. □
Since
P and
Q, defined in (
7), are nonsingular, we can easily show that for an arbitrary
matrix
X, there is an
matrix
Z such that
where
,
,
and
are
,
,
and
matrices, respectively. Now, we present our main results.
Theorem 1. Let be an idempotent matrix with rank r, and , defined in (8), an complex matrix. Then, X is a solution of YBME (1) if and only if , , and satisfy the following equations simultaneously: Proof. Notice that
,
and
, in a simple calculation lead to
and
. Thus,
Notice that
P and
Q are nonsingular; therefore,
X is a solution of YBME (
1) if and only if
,
and
satisfy (
9). □
In light of the previous result, note that the matrix
is completely arbitrary. In terms of Theorem 1, we can obtain all the solutions of YBME (
1) by solving (
9) for the matrices
(
). Denote the solution set of (
9) as
or equivalently,
In the following, we derive the set using the set .
(i) If
, then all the solutions of (
9) are
.
(ii) If
, then all the solutions of (
9) are
.
(iii) If , then we can assume that k where E, and F are two matrices of full column rank such that , .
Let
and
, where
are two
matrices whose columns form the basis of the null space of
and
, respectively. Then, we have
for the arbitrary
matrix
and arbitrary
matrix
. Thus, when
, we obtain the set of all the solutions of (
9) is
So, we find that the set
is
Corollary 1. Let be an idempotent matrix with rank r; then, all the solutions of YBME (1) can be expressed as in (8), where , and are arbitrary. For the commuting solutions of YBME (
1), we have an optimal result.
Theorem 2. Let , where U and V are two complex matrices of full column rank such that . Then, an complex matrix X is a commuting solution of YBME (1) if and only ifwhere E is any idempotent matrix, and F is any matrix. Proof. Sufficiency. Using Lemmas 2 and 3, we easily verify that when
E is an idempotent matrix; for arbitrary
F,
is a commuting solution of YBME (
1).
Necessity. Let
, where
Z is partitioned as in (
8). A simple calculation leads to
If
X commutes with
, then since
P,
Q are nonsingular, we obtain
and
. Thus,
where
is an
matrix,
is an
matrix.
To prove necessity, we still need to show that if
X in (
13) is a commuting solution of YBME (
1),
must be an idempotent matrix.
In fact, let
; we easily verify that
. Thus, it is clear that since
X is assumed to be a solution of YBME (
1),
needs to be an idempotent matrix. Let
and
, whereby the necessity is proved. □
Remark 1. Theorem 2 can be seen as a corollary of Theorem 1, and Lemmas 1 and 2 are corollaries of Theorem 2.
4. An Illustration Example
We provide an example to illustrate the process of constructing the solution set of YBME (
1).
Example 1. We easily obtain a spectral decomposition of A: with and . Since , we know that A is an idempotent matrix of rank 2.
A simple calculation leads to and .
Using Corollary 1, all the solutions of YBME (1) arewhere α is an arbitrary number, E is a matrix, and are 2-dimensional vectors. is in the set In the above set, are any two 2-dimensional vectors satisfying . are two 2-dimensional vectors satisfying . are numbers.
According to Theorem 2, all the commuting solutions of YBME (1) are as follows:where α is an arbitrary complex number, E is any idempotent matrix, that is, . Here, more specifically, means that , and are arbitrary 2-dimensional vectors satisfying .