Upper and Lower Bounds for Ranks of the Matrix Expression 𝑋−𝑋𝐴𝑋

Abstract

We consider the question of how to take 𝑋 such that the nonlinear matrix expression 𝑋−𝑋𝐴𝑋 attains its maximal and minimal possible ranks.

1. Introduction

Throughout this paper 𝐶^𝑚×𝑛 denotes the set of all 𝑚×𝑛 matrices over the complex field 𝐶. 𝐼_𝑘 denotes the identity matrix of order 𝑘 and 𝑂_𝑚×𝑛 is the 𝑚×𝑛 matrix of all zero entries (if no confusion occurs, we will drop the subscript). For a matrix 𝐴∈𝐶^𝑚×𝑛, 𝐴^∗ and 𝑟(𝐴) denote the conjugate transpose and the rank of the matrix 𝐴, respectively. (𝐴,𝐵) denotes a row block matrix consisting of 𝐴∈𝐶^𝑚×𝑛 and 𝐵∈𝐶^𝑚×𝑘.

Let 𝐴∈𝐶^𝑚×𝑛; a generalized inverse 𝑋 of 𝐴 is a matrix which satisfies some of the following four Penrose equations [1]: ()𝐴𝑋𝐴=𝐴,𝑋𝐴𝑋=𝑋,𝐴𝑋^∗=𝐴𝑋,(𝑋𝐴)^∗=𝑋𝐴. (1)

For a subset {𝑖,𝑗,…,𝑘} of the set {1,2,3,4}, the set of 𝑛×𝑚 matrices satisfying the equations (𝑖),(𝑗),…,(𝑘) from (1) is denoted by 𝐴{𝑖,𝑗,…,𝑘}. A matrix in 𝐴{𝑖,𝑗,…,𝑘} is called an {𝑖,𝑗,…,𝑘}-inverse of 𝐴 and is denoted by 𝐴^{(𝑖,𝑗,…,𝑘)}. For example, an 𝑛×𝑚 matrix 𝑋 of the set 𝐴{2} is called a {2}-inverse of 𝐴 and is denoted by 𝑋=𝐴⁽²⁾. The unique {1,2,3,4}-inverse of 𝐴 is denoted by 𝐴^†, which is called the Moore-Penrose inverse of 𝐴. For convenience, the symbols 𝐸_𝐴 and 𝐹_𝐴 stand for the two orthogonal projectors 𝐸_𝐴=𝐼_𝑚−𝐴𝐴^† and 𝐹_𝐴=𝐼_𝑛−𝐴^†𝐴. We refer the reader to [2–4] for basic results on generalized inverses.

In matrix theory and applications, there exists a nonlinear matrix expression that involves variable entries: 𝑃(𝑋)=𝑋−𝑋𝐴𝑋, (2) where 𝐴∈𝐶^𝑚×𝑚 is a given complex matrix and 𝑋∈𝐶^𝑚×𝑚 is a variable matrix. These nonlinear matrix expressions vary with respect to the choice of 𝑋. One of the fundamental problems for (2) is to determine the maximal and minimal possible ranks of the matrix expression 𝑃(𝑋) when 𝑋 is running over 𝐶^𝑚×𝑚. Since the rank of matrix is an integer between 0 and the minimum of row and column numbers of the matrix [5], then the maximal and minimal ranks of 𝑃(𝑋) can be attained for some 𝑋.

The investigation of extremal ranks of matrix expressions has many direct motivations in matrix analysis. For example, a matrix expression 𝐷−𝐴𝑋𝐵 of order 𝑛 is nonsingular if and only if the maximal rank of 𝐷−𝐴𝑋𝐵 with respect to 𝑋 is 𝑛; two consistent matrix equations 𝑋₁=𝑋₁𝐴𝑋₁ and 𝑋₂=𝑋₂𝐵𝑋₂ have a common solution if and only if the minimal rank of the difference 𝑋₁−𝑋₂ of their solutions is zero; a nonlinear matrix equation 𝑋=𝑋𝐴𝑋 is consistent if and only if the minimal rank of 𝑋−𝑋𝐴𝑋 with respect to 𝑋 is zero. From the definition of the {2}-inverse of a matrix, we know that the solution 𝑋 of the nonlinear matrix equation 𝑋=𝑋𝐴𝑋 is a {2}-inverse of matrix 𝐴; that is, using the minimal rank of 𝑋−𝑋𝐴𝑋, we can find out the general expression of the {2}-inverses of a matrix 𝐴, which is a matrix such that the nonlinear matrix expression 𝑋−𝑋𝐴𝑋 attains its minimal rank. In general, for any two matrix expressions 𝑃(𝑋₁,𝑋₂,…,𝑋_𝑚) and 𝑄(𝑌₁,𝑌₂,…,𝑌_𝑛) of the same size, there are 𝑋₁,𝑋₂,…,𝑋_𝑚 and 𝑌₁,𝑌₂,…,𝑌_𝑛 such that 𝑃(𝑋₁,𝑋₂,…,𝑋_𝑚)=𝑄(𝑌₁,𝑌₂,…,𝑌_𝑛) if and only if min_{𝑋1,𝑋2,…,𝑋𝑚;𝑌1,𝑌2,…,𝑌𝑛}𝑟𝑃𝑋₁,𝑋₂,…,𝑋_𝑚−𝑄𝑌₁,𝑌₂,…,𝑌_𝑛=0. (3)

These examples imply that the extremal ranks of matrix expressions have close links with many topics in matrix analysis and applications. Various statements on maximal and minimal ranks of matrix expressions are quite easy to understand for the people who know linear algebra. But the question now is how to give simple or closed forms for the extremal ranks of a matrix expression with respect to its variant matrices. The study on maximal and minimal ranks of matrix expression started in late 1980s. If want to know more about this question the reader can see [6–18].

The work in this paper includes two parts. First, in Section 2, we will consider how to choose a matrix 𝑋∈𝐶^𝑚×𝑚, such that 𝑋−𝑋𝐴𝑋 has the maximal possible rank. Second, in Section 3, we will determine the minimal rank of 𝑋−𝑋𝐴𝑋 and present a general expression of the {2}-inverses of matrix 𝐴∈𝐶^𝑚×𝑚.

In order to find the extremal ranks of the nonlinear matrix expression 𝑋−𝑋𝐴𝑋, we need the following lemmas, which will be used in this paper.

Lemma 1 (see [19]). Let 𝑃(𝑋₁,𝑋₂)=𝐴−𝐵₁𝑋₁𝐶₁−𝐵₂𝑋₂𝐶₂, where 𝐴∈𝐶^𝑚×𝑛, 𝐵₁∈𝐶^𝑚×𝑝1, 𝐵₂∈𝐶^𝑚×𝑝2, 𝐶₁∈𝐶^𝑞1×𝑛, and 𝐶₂∈𝐶^𝑞2×𝑛 are given matrices. Then for any variable matrices 𝑋₁∈𝐶^𝑝1×𝑞1 and 𝑋₂∈𝐶^𝑝2×𝑞2, one has max_𝑋1,𝑋2𝑟𝑃𝑋₁,𝑋₂=min𝑟𝐴,𝐵₁,𝐵₂𝐴,𝑟𝐶₁𝐶₂,𝑟𝐴𝐵₁𝐶₂𝑂,𝑟𝐴𝐵₂𝐶₁𝑂,min_𝑋1,𝑋2𝑟𝑃𝑋₁,𝑋₂𝐴=𝑟𝐶₁𝐶₂+𝑟𝐴,𝐵₁,𝐵₂+max𝑆₁,𝑆₂, (4)(5)
where 𝑆₁=𝑟𝐴𝐵₁𝐶₂𝑂−𝑟𝐴𝐵₁𝐵₂𝐶₂𝑂𝑂−𝑟𝐴𝐵₁𝐶₁𝑂𝐶₂𝑂,𝑆₂=𝑟𝐴𝐵₂𝐶₁𝑂−𝑟𝐴𝐵₁𝐵₂𝐶₁𝑂𝑂−𝑟𝐴𝐵₂𝐶₁𝑂𝐶₂𝑂. (6)

Lemma 2 (see [20]). Let 𝐴−𝐵𝑋𝐶 be a linear matrix expression over the complex field 𝐶, where 𝐴∈𝐶^𝑚×𝑛, 𝐵∈𝐶^𝑚×𝑘, and 𝐶∈𝐶^𝑙×𝑛 are given; 𝑋∈𝐶^𝑘×𝑙 is a variant matrix. Then the maximal rank of 𝐴−𝐵𝑋𝐶 with respect to 𝑋 is max_𝑋𝑟()𝑟()𝐴𝐶𝐴−𝐵𝑋𝐶=min𝐴,𝐵,𝑟, (7) the general expression of 𝑋 satisfying (7) is 𝑋=𝐸_𝐴2𝐵^†𝐸_𝐴2𝐴𝐹_𝐴1𝐶𝐹_𝐴1^†+𝑈, (8) where 𝐴₁=𝐸_𝐵𝐴=(𝐼_𝑚−𝐵𝐵^†)𝐴, 𝐴₂=𝐴𝐹_𝐶=𝐴(𝐼_𝑛−𝐶^†𝐶)𝐴, 𝐸_𝐴2=𝐼_𝑚−𝐴₂𝐴^†₂, and 𝐹_𝐴1=𝐼_𝑛−𝐴^†₁𝐴₁, and the matrix 𝑈∈𝐶^𝑘×𝑙 is chosen such that 𝑟𝐸_𝐴2𝐵𝑈𝐶𝐹_𝐴1=min𝑟𝐸_𝐴2𝐵,𝑟𝐶𝐹_𝐴1. (9)

Lemma 3 (see [21]). Let 𝐴∈𝐶^𝑚×𝑛 and 𝐵∈𝐶^𝑚×𝑘. Then (𝐴,𝐵)^†=𝐼_𝑛+𝑇𝑇^∗−1𝐴^†−𝐴^†𝐵𝐶^†𝑇^∗𝐼_𝑛+𝑇𝑇^∗−1𝐴^†−𝐴^†𝐵𝐶^†+𝐶^†, (10) where 𝐶=(𝐼_𝑚−𝐴𝐴^†)𝐵, 𝑇=𝐴^†𝐵(𝐼_𝑘−𝐶^†𝐶), and (𝐼_𝑛+𝑇𝑇^∗)⁻¹=𝐼_𝑛−𝑇(𝐼_𝑘+𝑇^∗𝑇)⁻¹𝑇^∗.

Lemma 4 (see [22]). Let 𝐴∈𝐶^𝑚×𝑛, 𝐵∈𝐶^𝑚×𝑘, and 𝐶∈𝐶^𝑙×𝑛. Then 𝑟(𝐴,𝐵)=𝑟(𝐴)+𝑟𝐸_𝐴𝐵=𝑟(𝐵)+𝑟𝐸_𝐵𝐴,𝑟𝐴𝐶=𝑟(𝐴)+𝑟𝐶𝐹_𝐴=𝑟(𝐶)+𝑟𝐴𝐹_𝐶,𝑟(𝐴+𝐵)≤𝑟(𝐴)+𝑟(𝐵), (11) where 𝐸_𝐴=𝐼_𝑚−𝐴𝐴^†, 𝐸_𝐵=𝐼_𝑚−𝐵𝐵^†, 𝐹_𝐴=𝐼_𝑛−𝐴^†𝐴 and 𝐹_𝐶=𝐼_𝑛−𝐶^†𝐶.

2. The Maximal Rank of 𝑋−𝑋𝐴𝑋 with respect to 𝑋

Let 𝐴∈𝐶^𝑚×𝑚 be a given matrix; in this section, we will present the maximal rank of the nonlinear matrix expression 𝑋−𝑋𝐴𝑋, with respect to the variable matrix 𝑋∈𝐶^𝑚×𝑚. The relative results are included in the following three lemmas.

Lemma 5. Let 𝐴∈𝐶^𝑚×𝑚, and 𝐼=𝐼_𝑚 denotes the identity matrix of order 𝑚. Then 𝐼𝐼𝑂𝐴^†=𝐼+𝐹_𝐴⁻¹−𝐴^†𝐹_𝐴𝐼+𝐹_𝐴⁻¹𝐴^†. (12)

Proof. By Lemma 3 with 𝑀=^𝐼_𝑂 and 𝑁=^𝐼_𝐴, we have 𝐼𝐼𝑂𝐴^†=(𝑀,𝑁)^†=𝐼+𝑇𝑇^∗−1𝑀^†−𝑀^†𝑁𝐾^†𝑇^∗𝐼+𝑇𝑇^∗−1𝑀^†−𝑀^†𝑁𝐾^†+𝐾^†, (13) where 𝑀^†=(𝐼,𝑂),𝐾=𝐼−𝑀𝑀^†𝑂𝐴,𝐾𝑁=_†=𝑂,𝐴^†,𝑀^†−𝑀^†𝑁𝐾^†=𝐼,𝐴^†,𝑇=𝑀^†𝑁𝐼−𝐾^†𝐾=𝐹_𝐴=𝐼−𝐴^†𝐴. (14)(15)
Combining (13), (14) with (15), we have 𝐼𝐼𝑂𝐴^†=𝐼+𝐹_𝐴⁻¹−𝐼+𝐹_𝐴⁻¹𝐴^†𝐹_𝐴𝐼+𝐹_𝐴⁻¹−𝐹_𝐴𝐼+𝐹_𝐴⁻¹𝐴^†+𝐴^†=𝐼+𝐹_𝐴⁻¹−𝐴^†𝐹_𝐴𝐼+𝐹_𝐴⁻¹𝐴^†. (16) The second equality holds as 𝐼+𝐹_𝐴⁻¹=𝐼−𝐹_𝐴𝐼+𝐹_𝐴⁻¹𝐹_𝐴,𝐹_𝐴𝐴^†=𝐼−𝐴^†𝐴𝐴^†=𝑂. (17)

Lemma 6. Let 𝐴∈𝐶^𝑚×𝑚, and 𝐼=𝐼_𝑚 denotes the identity matrix of order 𝑚. Then max_𝑈𝑟−𝑈(𝐴).𝑂𝑂𝑂𝐴𝐼𝐼𝑂𝐴𝑂−𝐴−𝐼𝑂=𝑚+𝑟 (18)
The general expression of 𝑈 satisfying (18) is 𝐴𝑈=^†+𝑉₁𝑉₂−𝐴^†+𝑉₃𝑉₄, (19) where 𝑉₁,𝑉₂,𝑉₃,𝑉₄∈𝐶^𝑚×𝑚 are chosen such that 𝑟𝐼𝐼𝑉𝑂𝐴¹𝑉₂𝑉₃𝑉₄(𝐴).𝑂−𝐴−𝐼𝑂=𝑚+𝑟 (20)

Proof. By Lemma 2 with 𝑃=^𝑂𝑂_𝑂𝐴, 𝑄=^𝐼𝐼_𝑂𝐴, and 𝑊=^𝑂−𝐴_−𝐼𝑂, we have max_𝑈𝑟−𝑈𝑂𝑂𝑂𝐴𝐼𝐼𝑂𝐴𝑂−𝐴−𝐼𝑂=max_𝑈𝑟()𝑟()𝑃𝑊𝑟(𝐴).𝑃−𝑄𝑈𝑊=min𝑃,𝑄,𝑟=min𝑂𝑂𝐼𝐼𝑂𝐴𝑂𝐴,𝑟𝑂𝑂𝑂𝐴𝑂−𝐴−𝐼𝑂=𝑚+𝑟 (21)
From Lemmas 2 and 5, we have the general expression of 𝑈 satisfying (21) as 𝑈=𝐸_𝐴2𝑄^†𝐸_𝐴2𝑃𝐹_𝐴1𝑊𝐹_𝐴1^†+𝑉, (22) where 𝑉𝑉=¹𝑉₂𝑉₃𝑉₄,𝑄^†=𝐼+𝐹_𝐴⁻¹−𝐴^†𝐹_𝐴𝐼+𝐹_𝐴⁻¹𝐴^†,𝑊^†=𝑂−𝐼−𝐴^†𝑂,𝐴₁=𝐸_𝑄𝑃=𝐼−𝑄𝑄^†𝑃=𝐼−𝐼𝐼𝑂𝐴𝐼+𝐹_𝐴⁻¹−𝐴^†𝐹_𝐴𝐼+𝐹_𝐴⁻¹𝐴^†=𝑂𝑂𝑂𝐴𝑂𝑂𝑂𝐸_𝐴𝑂𝑂𝑂𝐴=𝑂,𝐴₂=𝑃𝐹_𝑊=𝑃𝐼−𝑊^†𝑊=𝑂𝑂𝑂𝐴𝐼−𝑂−𝐼−𝐴^†𝑂=𝑂−𝐴−𝐼𝑂𝑂𝑂𝑂𝐴𝑂𝑂𝑂𝐹_𝐴=𝑂. (23)(24)
Combining (22), (23) with (24), we have 𝐸_𝐴2=𝐼, 𝐹_𝐴1=𝐼 and 𝑈=𝑄^†𝑃𝑊^†=+𝑉𝐼+𝐹_𝐴⁻¹−𝐴^†𝐹_𝐴𝐼+𝐹_𝐴⁻¹𝐴^†𝑂𝑂𝑂𝐴𝑂−𝐼−𝐴^†𝑂+𝑉₁𝑉₂𝑉₃𝑉₄=𝐴^†+𝑉₁𝑉₂−𝐴^†+𝑉₃𝑉₄, (25) where 𝑉₁,𝑉₂,𝑉₃,𝑉₄∈𝐶^𝑚×𝑚 are chosen such that 𝑟𝐸_𝐴2𝑄𝑉₁𝑉₂𝑉₃𝑉₄𝑊𝐹_𝐴1𝑉=𝑟𝐼𝐼𝑂𝐴¹𝑉₂𝑉₃𝑉₄𝑟𝐸𝑂−𝐴−𝐼𝑂=min_𝐴2𝑄,𝑟𝑊𝐹_𝐴1{𝑟(𝑄)(𝑊(𝐴).=min,𝑟)}=𝑚+𝑟 (26)

Lemma 7. Let 𝐴∈𝐶^𝑚×𝑚. Then there exist some matrices 𝑉∈𝐶^𝑚×𝑚, such that max_𝑉𝑟𝐴^†+𝑉−𝐴^†+𝑉𝐴𝐴^†+𝑉=𝑚. (27)

Proof. Applying Lemma 4, we have 𝑟𝐴^†+𝑉−𝐴^†+𝑉𝐴𝐴^†𝐴+𝑉=𝑟^†−𝐴^†𝐴𝐴^†+𝑉−𝑉𝐴^†𝐴−𝐴^†𝐴𝐴𝑉−𝑉𝐴𝑉≤𝑟^†−𝐴^†𝐴𝐴^†+𝑟𝑉−𝑉𝐴^†𝐴−𝐴^†𝐴𝑉−𝑉𝐴𝑉=0+𝑟𝑉−𝑉𝐴^†𝐴−𝐴^†𝐴𝑉−𝑉𝐴𝑉=𝑟𝑉−𝑉𝐴𝐴^†−𝐴^†(𝐴)−𝑂𝑉𝐴𝑉𝑉𝐴𝐴𝑉𝐴−𝑟=𝑟𝑂𝑂𝑂𝐴−𝐼𝐼−𝐴𝐴^†−𝐴,𝐴_†𝐴𝑉()(𝐴).−𝐴𝐼,𝑂−𝑟 (28)
From (27) and (28), we have max_𝑉𝑟𝐴^†+𝑉−𝐴^†+𝑉𝐴𝐴^†+𝑉=max_𝑉𝑟−𝑂𝑉𝑂𝑂𝑂𝐴−𝐼𝐼−𝐴𝐴^†−𝐴,𝐴_†𝐴𝑉()(𝐴).−𝐴𝐼,𝑂−𝑟 (29)By Lemma 1, we have max_𝑉𝑟−𝑂𝑉𝑂𝑂𝑂𝐴−𝐼𝐼−𝐴𝐴^†−𝐴,𝐴_†𝐴𝑉()𝑟−𝐴𝐼,𝑂=min𝑂𝑂−𝐼𝐴^†𝐴𝑂𝐴𝑂𝐴,𝑟𝑂𝑂𝑂𝐴𝐼−𝐴𝐴^†𝐴,𝑟𝐼𝑂𝑂𝑂−𝐼𝑂𝐴𝑂𝐼𝑂𝑂,𝑟𝑂𝑂𝐴^†𝐴𝑂𝐴−𝐴𝐼−𝐴𝐴^†{(𝐴)(𝐴)(𝐴)(𝐴(𝐴).𝐴𝑂=min𝑚+𝑟,𝑚+𝑟,2𝑚+𝑟,𝑚+𝑟)}=𝑚+𝑟 (30)
Combining (29) with (30), we get the result (27).

According to Lemmas 5, 6, and 7, we immediately obtain the following theorem.

Theorem 8. Let 𝐴∈𝐶^𝑚×𝑚 be a given matrix, and 𝑋∈𝐶^𝑚×𝑚 is a variant matrix. Then max_𝑋𝑟(𝑋−𝑋𝐴𝑋)=𝑚. (31)
In consequence, (1)there always exists 𝑋∈𝐶^𝑚×𝑚, such that 𝑋−𝑋𝐴𝑋 is nonsingular;(2)the matrices 𝑋 satisfying (31) are given by 𝑋=𝐴^†+𝑉 and 𝑉∈𝐶^𝑚×𝑚 is the same as in (27).

Proof. First describe a special congruence transformation for a block matrix, which reduces the calculation of the maximal rank of 𝑋−𝑋𝐴𝑋: max_𝑋𝑟(𝑋−𝑋𝐴𝑋)=max_𝑋(𝐴)𝑋𝑋𝐴𝐴𝑋𝐴−𝑟=max_𝑋𝑟−𝑋()−𝐼𝑂𝑋𝑂(𝐴).𝑂𝑂𝑂𝐴−𝐼−𝐴𝐼,𝑂−𝐴−𝑟 (32)
By Lemma 1, we have max_𝑋𝑟−𝑋()−𝐼𝑂𝑋𝑂𝑟,𝑟{(𝐴)(𝐴)(𝐴)(𝐴(𝐴).𝑂𝑂𝑂𝐴−𝐼−𝐴𝐼,𝑂−𝐴=min𝑂𝑂−𝐼𝐼𝑂𝐴−𝐴𝑂,𝑟𝑂𝑂𝑂𝐴𝐼𝑂𝑂−𝐴𝑂𝑂−𝐼𝑂𝐴−𝐴𝑂−𝐴𝑂,𝑟𝑂𝑂𝐼𝑂𝐴𝑂𝐼𝑂𝑂=min𝑚+𝑟,𝑚+𝑟,𝑚+𝑟,2𝑚+𝑟)}=𝑚+𝑟 (33)Combining (32) with (33), we have max_𝑋𝑟(𝑋−𝑋𝐴𝑋)=𝑚. (34)
On the other hand, from another special congruence transformation for a block matrix, we have max_𝑋𝑟(𝑋−𝑋𝐴𝑋)=max_𝑋(𝐴)𝑋𝑋𝐴𝐴𝑋𝐴−𝑟=max_𝑋𝑟−(𝐴).𝑂𝑂𝑂𝐴𝐼𝐼𝑂𝐴𝑋𝑂𝑂𝑋𝑂−𝐴−𝐼𝑂−𝑟 (35)
Combining formula (35) with Lemma 6, we have max_𝑋𝑟(𝑋−𝑋𝐴𝑋)=max_𝑋𝑟−(𝐴)𝑟(𝐴)(𝐴)(𝐴)𝑂𝑂𝑂𝐴𝐼𝐼𝑂𝐴𝑋𝑂𝑂𝑋𝑂−𝐴−𝐼𝑂−𝑟=min𝑂𝑂𝐼𝐼𝑂𝐴𝑂𝐴,𝑟𝑂𝑂𝑂𝐴𝑂−𝐴−𝐼𝑂−𝑟=𝑚+𝑟−𝑟=𝑚. (36) That is, theres always exist 𝑋∈𝐶^𝑚×𝑚, such that 𝑋−𝑋𝐴𝑋 is nonsingular.
From the results in Lemmas 6 and 7, we obtain that there always exist 𝑋=𝐴^†+𝑉, such that max_𝑋𝑟(𝑋−𝑋𝐴𝑋)=max_𝑉𝑟𝐴^†−𝐴+𝑉_†+𝑉𝐴𝐴^†+𝑉=𝑚. (37)

3. The Minimal Rank of 𝑋−𝑋𝐴𝑋 with respect to 𝑋

In this section, we will present the minimal rank of the nonlinear matrix expression 𝑋−𝑋𝐴𝑋. Moreover, we will consider how to choose a matrix 𝑋, such that 𝑋−𝑋𝐴𝑋 has the minimal possible rank.

Theorem 9. Let 𝐴∈𝐶^𝑚×𝑚 be a given matrix, and 𝑋∈𝐶^𝑚×𝑚 is a variant matrix. Then min_𝑋𝑟(𝑋−𝑋𝐴𝑋)=0. (38) In consequence, there exists 𝑋∈𝐶^𝑚×𝑚, such that the nonlinear matrix equation 𝑋=𝑋𝐴𝑋 is consistent.

Proof. By formula (5) in Lemma 1, we have min_𝑋𝑟(𝑋−𝑋𝐴𝑋)=min_𝑋(𝐴)𝑋𝑋𝐴𝐴𝑋𝐴−𝑟=max_𝑋𝑟−𝑋()−𝐼𝑂𝑋𝑂(𝐴)𝑆𝑂𝑂𝑂𝐴−𝐼−𝐴𝐼,𝑂−𝐴−𝑟=𝑟𝑂𝑂𝑂𝐴𝑂−𝐴−𝐼𝑂+𝑟𝑂𝑂𝐼𝐼𝑂𝐴𝑂𝐴+max₁,𝑆₂(𝐴)(𝐴)𝑆−𝑟=2𝑚+𝑟+max₁,𝑆₂, (39) where 𝑆₁(𝐴),=𝑟𝑂𝑂𝐼𝑂𝐴𝑂−𝐼𝑂𝑂−𝑟𝑂𝑂𝐼𝐼𝑂𝐴𝑂𝐴−𝐼𝑂𝑂𝑂−𝑟𝑂𝑂𝐼𝑂𝐴𝑂𝑂−𝐴𝑂−𝐼𝑂𝑂=−2𝑚−𝑟𝑆₂(𝐴).=𝑟𝑂𝑂𝐼𝑂𝐴𝐴𝑂−𝐴𝑂−𝑟𝑂𝑂𝐼𝐼𝑂𝐴𝑂𝐴𝑂−𝐴𝑂𝑂−𝑟𝑂𝑂𝐼𝑂𝐴𝐴𝑂−𝐴𝑂−𝐼𝑂𝑂=−2𝑚−𝑟 (40)(41) Combining (39), (40) with (41), we have min_𝑋𝑟(𝑋−𝑋𝐴𝑋)=0. (42)

Corollary 10. Let 𝐴∈𝐶^𝑚×𝑚 be a given matrix. Then the matrix 𝑋∈𝐶^𝑚×𝑚 satisfying the matrix equation 𝑋=𝑋𝐴𝑋 is given by 𝑋=𝐴^†+𝐹_𝐴𝑉𝐴𝐴^†+𝑊𝐸_𝐴, (43) where 𝑉,𝑊∈𝐶^𝑚×𝑚 are two variant matrices.

Proof. Putting 𝑋=(𝐴^†+𝐹_𝐴𝑉)𝐴(𝐴^†+𝑊𝐸_𝐴) into 𝑋−𝑋𝐴𝑋 yields =𝐴𝑋−𝑋𝐴𝑋_†+𝐹_𝐴𝑉𝐴𝐴^†+𝑊𝐸_𝐴−𝐴^†+𝐹_𝐴𝑉𝐴𝐴^†𝐴𝐴^†+𝐹_𝐴𝑉𝐴𝐴^†+𝑊𝐸_𝐴=𝐴^†+𝐹_𝐴𝑉𝐴𝐴^†+𝑊𝐸_𝐴−𝐴^†+𝐹_𝐴𝑉𝐴𝐴^†𝐴𝐴^†+𝑊𝐸_𝐴=𝑂. (44)

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank Professor Changbum Chun and the anonymous referees for their very detailed comments and constructive suggestions, which greatly improved the presentation of this paper. This work was supported by the NSFC (Grant no. 11301397), the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (Grant no. 2012LYM-0126), and the Basic Theory and Scientific Research of Science and Technology Project of Jiangmen City, China, 2013.