Iterative Algorithm for Solving a Class of Quaternion Matrix Equation over the Generalized (𝑃,𝑄)-Reflexive Matrices

Abstract

The matrix equation ∑^𝑢_𝑙=1𝐴_𝑙𝑋𝐵_𝑙+∑^𝑣_𝑠=1𝐶_𝑠𝑋^𝑇𝐷_𝑠=𝐹, which includes some frequently investigated matrix equations as its special cases, plays important roles in the system theory. In this paper, we propose an iterative algorithm for solving the quaternion matrix equation ∑^𝑢_𝑙=1𝐴_𝑙𝑋𝐵_𝑙+∑^𝑣_𝑠=1𝐶_𝑠𝑋^𝑇𝐷_𝑠=𝐹 over generalized (𝑃,𝑄)-reflexive matrices. The proposed iterative algorithm automatically determines the solvability of the quaternion matrix equation over generalized (𝑃,𝑄)-reflexive matrices. When the matrix equation is consistent over generalized (𝑃,𝑄)-reflexive matrices, the sequence {𝑋(𝑘)} generated by the introduced algorithm converges to a generalized (𝑃,𝑄)-reflexive solution of the quaternion matrix equation. And the sequence {𝑋(𝑘)} converges to the least Frobenius norm generalized (𝑃,𝑄)-reflexive solution of the quaternion matrix equation when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate generalized (𝑃,𝑄)-reflexive solution for a given generalized (𝑃,𝑄)-reflexive matrix 𝑋₀ can be derived. The numerical results indicate that the iterative algorithm is quite efficient.

1. Introduction and Notations

Quaternion matrix equation is one of the topics of very active research in matrix theory and applications, and some important results have been developed. For example, Kyrchei [1] derived the explicit representation formulas for the minimum norm least squares solutions of quaternion matrix equations 𝐴𝑋=𝐵,  𝑋𝐴=𝐵,  𝐴𝑋𝐵=𝐷. Yuan and Liao [2] derived the expressions of the least squares (𝑗-self-conjugate) solution with the least norm of the quaternion matrix equation 𝑋−𝐴𝑋𝐵=𝐶. Song et al. [3] obtained the expressions of the explicit solutions of quaternion matrix equations 𝑋𝐹−𝐴𝑋=𝐵𝑌 and 𝑋𝐹−𝐴𝑋=𝐵𝑌. He and Wang [4] studied the necessary and sufficient conditions for the existence of a solution to the quaternion matrix equation 𝐴₁𝑋+(𝐴₁𝑋)^𝜂∗+𝐵₁𝑌(𝐵₁)^𝜂∗+𝐶₁𝑍(𝐶₁)^𝜂∗=𝐷₁ and derived a general solution when the equation is solvable.

The definition of generalized (𝑃,𝑄)-reflexive matrix can be found in [5]. A complex matrix 𝐴 is called (𝑃,𝑄)-generalized reflexive (generalized anti-reflexive) if 𝐴=𝑃𝐴𝑄  (𝐴=−𝑃𝐴𝑄), where 𝑃 and 𝑄 are two generalized reflection matrices; that is, 𝑃=𝑃^𝐻=𝑃⁻¹ and 𝑄=𝑄^𝐻=𝑄⁻¹. The generalized (𝑃,𝑄)-reflexive matrices have been widely used in engineering and scientific computations [5–7]. Throughout, we use ℍ_𝑟^𝑚×𝑛(𝑃,𝑄) to denote the set of  𝑚×𝑛  generalized (𝑃,𝑄)-reflexive quaternion matrices.

The iterative method is a very important method to solve matrix equations. Peng et al. [8, 9] constructed iteration methods to solve the symmetric and reflexive solutions of matrix equations 𝐴₁𝑋𝐵₁=𝐶₁,  𝐴₂𝑋𝐵₂=𝐶₂. Ding and Chen [10, 11] developed iterative methods for solving a wide variety of matrix equations, based on the hierarchical identification principle [12]. Peng [13–15] presented several efficient iterative methods to solve the constrained least squares solutions of linear matrix equations 𝐴𝑋𝐵=𝐶 and 𝐴𝑋𝐵+𝐶𝑌𝐷=𝐸, by using Paige's algorithm [16] as the frame method. Wang et al. [17] proposed iterative algorithms for solving the matrix equation 𝐴𝑋𝐵+𝐶𝑋^𝑇𝐷=𝐸. Xie et al. [18] presented a gradient based iterative algorithm for solving ^𝑢_𝑙=1𝐴_𝑙𝑋𝐵_𝑙+^𝑣_𝑠=1𝐶_𝑠𝑋^𝑇𝐷_𝑠=𝐹. (1) In [19], Li et al. proposed iterative algorithms to solve the minimal norm least squares solution to (1). Duan et al. [20–24] proposed iterative algorithms for the (Hermitian) positive definite solutions of some nonlinear matrix equations. Dehghan and Hajarian constructed iterative algorithms to solve several linear matrix equations over (anti-)reflexive [25, 26], generalized centrosymmetric [27, 28], and generalized bisymmetric [29, 30] matrices. Recently, Wu et al. [31–35] proposed iterative algorithms for solving various complex matrix equations. Wang et al. [36] derived an iterative method for finding the minimum-norm solution of the QLS problem in quaternionic quantum theory.

However, to our best knowledge, the generalized (𝑃,𝑄)-reflexive solution of (1) over the quaternion algebra ℍ has not been considered so far. Due to the noncommutativity of ℍ, some well-known equalities for complex and real matrices no longer hold for quaternion matrices, which make the study of quaternion matrix equation more complex than that of real and complex equation. Motivated by the work mentioned previously and keeping the interests and wide applications of quaternion matrices in view (e.g., [37–47]), we, in this paper, consider an iterative method for the following two problems.

Problem 1. For given matrices 𝐴_𝑙, 𝐶_𝑠∈ℍ^𝑝×𝑛, 𝐵_𝑙, 𝐷_𝑠∈ℍ^𝑛×𝑞, 𝑙=1,2,…,𝑢, 𝑠=1,2,…,𝑣, 𝐹∈ℍ^𝑝×𝑞, and the generalized reflection matrices 𝑃∈ℍ^𝑛×𝑛, 𝑄∈ℍ^𝑛×𝑛, find that 𝑋∈ℍ_𝑟^𝑛×𝑛(𝑃,𝑄), such that ^𝑢_𝑙=1𝐴_𝑙𝑋𝐵_𝑙+^𝑣_𝑠=1𝐶_𝑠𝑋^𝑇𝐷_𝑠=𝐹. (2)

Problem 2. When Problem 1 is consistent, let its solution set be denoted by 𝑆_𝑋. For a given (𝑃,𝑄)-reflexive matrix 𝑋₀∈ℍ_𝑟^𝑛×𝑛(𝑃,𝑄), find 𝑋∈ℍ_𝑟^𝑛×𝑛(𝑃,𝑄), such that 𝑋−𝑋₀=min_{𝑋∈𝑆𝑋}𝑋−𝑋₀. (3)

The matrix equation (2) plays important roles in the system theory [48–50]. Moreover, (2) obviously includes the matrix equations 𝐴𝑋𝐵+𝐶𝑋^𝑇𝐷=𝐹, 𝐴𝑋+𝑋^𝑇𝐷=𝐹, and 𝐴𝑋𝐵=𝐶 as special cases, which have been investigated in [17, 28, 36, 51–53].

Throughout this paper, ℝ^𝑚×𝑛 and ℍ^𝑚×𝑛 represent the set of all 𝑚×𝑛 real matrices and the set of all 𝑚×𝑛 matrices over the quaternion algebra ℍ={𝑎₁+𝑎₂𝑖+𝑎₃𝑗+𝑎₄𝑘∣𝑖²=𝑗²=𝑘²=𝑖𝑗𝑘=−1,𝑎₁,𝑎₂,𝑎₃,𝑎₄∈ℝ}. We use 𝐴^𝐻,  𝐴^𝑇,  𝐴,  tr(𝐴),  𝑅(𝐴),  Re(𝐴), and vec(𝐴) to denote the conjugate transpose, the transpose, the conjugate, the trace, the column space, the real part, and the 𝑚𝑛×1 vector formed by the vertical concatenation of the respective columns of a matrix 𝐴; respectively. We use ‖𝐴‖ to denote the Frobenius norm of 𝐴, that is, ‖𝐴‖=tr(𝐴^𝐻𝐴). The Kronecker matrix product and Hadamard matrix product of the matrices 𝐴 and 𝐵 are denoted by 𝐴⊗𝐵 and 𝐴⊙𝐵. We use 𝐼 to denote the identity matrix with the appropriate size.

2. Preliminaries

In this section, we provide some results which will play important roles in this paper.

Firstly, we recall the definition of real inner product space. Then we define a real inner product in the space ℍ^𝑚×𝑛 over the field ℝ.

Definition 3 (see [54]). A real inner product space is a vector space 𝑉 over the real field ℝ together with an inner product defined by a mapping ⟨⋅,⋅⟩∶𝑉×𝑉⟶ℝ (4) satisfying the following three axioms for all vectors 𝑥,𝑦,𝑧∈𝑉 and all scalars 𝑎∈ℝ.(1)Symmetry: ⟨𝑥,𝑦⟩=⟨𝑦,𝑥⟩;(2)Linearity in the first argument: ⟨𝑎𝑥,𝑦⟩=𝑎⟨𝑥,𝑦⟩,⟨𝑥+𝑦,𝑧⟩=⟨𝑥,𝑧⟩+⟨𝑦,𝑧⟩;(3)Positive definiteness: ⟨𝑥,𝑥⟩>0 for all 𝑥≠0.

In [33–35], the following real inner product was presented to solve some complex matrix equations∶⟨𝐴,𝐵⟩=Retr𝐵^𝐻𝐴, (5) for 𝐴,𝐵∈ℂ^𝑚×𝑛. It is easy to verify that if we let 𝐴,𝐵∈ℍ^𝑚×𝑛, (5) also defines a real inner product in ℍ^𝑚×𝑛 over ℝ. We denote this real inner product space by (ℍ^𝑚×𝑛,ℝ,⟨⋅,⋅⟩). From the equalities ⟨𝐴,𝐴⟩=Retr𝐴^𝐻𝐴=tr𝐴^𝐻𝐴=‖𝐴‖,𝐴∈ℍ^𝑚×𝑛, (6) we know that the induced matrix norm by the introduced inner product ⟨⋅,⋅⟩ is exactly the Frobenius norm. For convenience, we still use ‖⋅‖ to denote the induced matrix norm. Let 𝐸_𝑖𝑗 denote the 𝑚×𝑛 matrix whose (𝑖,𝑗) entry is 1, and the other elements are zeros. In inner product space (ℍ^𝑚×𝑛,ℝ,⟨⋅,⋅⟩), it is easy to verify that 𝐸_𝑖𝑗, 𝐸_𝑖𝑗𝑖, 𝐸_𝑖𝑗𝑗,  𝐸_𝑖𝑗𝑘,  𝑖=1,2,…𝑚,  𝑗=1,2,…𝑛, is an orthonormal basis, which reveals that the dimension of the inner product space (ℍ^𝑚×𝑛,ℝ,⟨⋅,⋅⟩) is 4𝑚𝑛.

Next, we introduce the well-known real representation of a quaternion matrix. For an arbitrary quaternion matrix 𝑀=𝑀₁+𝑀₂𝑖+𝑀₃𝑗+𝑀₄𝑘∈ℍ^𝑚×𝑛, a map 𝜙(⋅), from ℍ^𝑚×𝑛 to ℝ^4𝑚×4𝑛, can be defined as 𝜙(𝑀)=𝑀₁−𝑀₂−𝑀₃−𝑀₄𝑀₂𝑀₁−𝑀₄𝑀₃𝑀₃𝑀₄𝑀₁−𝑀₂𝑀₄−𝑀₃𝑀₂𝑀₁. (7) Let 𝑀 and 𝑁 be two arbitrary quaternion matrices with appropriate size. We know that 𝜙(⋅) satisfies the following properties: 𝑀=𝑁⟺𝜙(𝑀)=𝜙(𝑁),𝜙()(𝑀)(𝑁),𝜙()(𝑀)𝜙(𝑁),𝜙()(𝑀)𝑀+𝑁=𝜙+𝜙𝑀𝑁=𝜙𝑘𝑀=𝑘𝜙,𝑘∈ℝ,𝜙𝑀^𝐻=𝜙^𝑇(𝑀). (8)(9)(10) It follows a simple verification that 𝜙(⋅) also satisfies the following two properties which is useful for some deduction in this paper: 𝜙(𝑀)=2‖𝑀‖,𝜙𝑀^𝑇=2𝜙^𝑇(𝑀)⊙𝑊−𝜙^𝑇(𝑀), (11)(12) where 𝑊=𝐸0000𝐸0000𝐸0000𝐸 (13) and 𝐸 is a matrix whose elements are all equal to one with the same size as 𝑀.

Finally, we recall some results about the commutation matrix. A commutation matrix 𝑃(𝑚,𝑛) is a 𝑚𝑛×𝑚𝑛 matrix which has the following explicit form: 𝑃(𝑚,𝑛)=^𝑚𝑛_𝑖=1𝑗=1𝐸_𝑖𝑗⊗𝐸^𝑇_𝑖𝑗=𝐸^𝑇_{𝑖𝑗𝑖=1,…,𝑚𝑗=1,…,𝑛}. (14) Moreover, 𝑃(𝑚,𝑛) is a permutation matrix and 𝑃(𝑚,𝑛)=𝑃^𝑇(𝑛,𝑚)=𝑃⁻¹(𝑛,𝑚). We have the following lemmas on the commutation matrix.

Lemma 4 (see [55]). Let 𝐴 be an 𝑚×𝑛 matrix. There is a commutation matrix 𝑃(𝑚,𝑛) such that vec𝐴^𝑇=𝑃(𝑚,𝑛)vec(𝐴). (15)

Lemma 5 (see [55]). Let 𝐴 be an 𝑚×𝑛 matrix and 𝐵 a 𝑝×𝑞 matrix. There exist two commutation matrices 𝑃(𝑚,𝑝) and 𝑃(𝑛,𝑞) such that 𝐵⊗𝐴=𝑃^𝑇𝑚,𝑝(𝐴⊗𝐵)𝑃𝑛,𝑞. (16)

3. An Iterative Algorithm for Solving Problem 1

In this section, we firstly construct an algorithm for solving Problem 1. Then we show that if the problem is consistent, the sequence {𝑋(𝑘)} generated by the introduced algorithm converges to a solution of Problem 1 within at most 4𝑝𝑞 iteration steps.

Algorithm 6. (1) Input matrices 𝐴_𝑙,𝐶_𝑠∈ℍ^𝑝×𝑛,  𝐵_𝑙,  𝐷_𝑠∈ℍ^𝑛×𝑞,  𝑙=1,2,…,𝑢,  𝑠=1,2,…,𝑣,  𝐹∈ℍ^𝑝×𝑞,   generalized reflection matrices 𝑃∈ℍ^𝑛×𝑛,  𝑄∈ℍ^𝑛×𝑛,  𝑋(1)∈ℍ_𝑟^𝑛×𝑛(𝑃,𝑄).
(2) Calculate 𝑅(1)=𝐹−^𝑢_𝑙=1𝐴_𝑙𝑋(1)𝐵_𝑙−^𝑣_𝑠=1𝐶_𝑠𝑋^𝑇(1)𝐷_𝑠;𝑆(1)=12^𝑢_𝑙=1𝐴^𝐻_𝑙𝑅(1)𝐵^𝐻_𝑙+^𝑣_𝑠=1𝐶^𝐻_𝑠𝑅(1)𝐷^𝐻_𝑠^𝑇+^𝑢_𝑙=1𝑃𝐴^𝐻_𝑙𝑅(1)𝐵^𝐻_𝑙𝑄+^𝑣_𝑠=1𝑃𝐶^𝐻_𝑠𝑅(1)𝐷^𝐻_𝑠^𝑇𝑄;𝑇(1)=𝑆(1)𝑘∶=1. (17)
(3) If 𝑅(𝑘)=0, or 𝑅(𝑘)≠0 and 𝑇(𝑘)=0, stop; otherwise 𝑘∶=𝑘+1.
(4) Calculate 𝑋(𝑘)()+‖‖=𝑋𝑘−1𝑅(𝑘−1)²‖𝑇(𝑘−1)‖²𝑇();𝑅(𝑘)()−‖𝑅(𝑘−1=𝑅𝑘−1𝑘−1)‖²‖𝑇(𝑘−1)‖^2𝑢_𝑙=1𝐴_𝑙𝑇(𝑘−1)𝐵_𝑙+^𝑣_𝑠=1𝐶_𝑠𝑇^𝑇(𝑘−1)𝐷_𝑠;𝑆(𝑘)=12^𝑢_𝑙=1𝐴^𝐻_𝑙𝑅(𝑘)𝐵^𝐻_𝑙+^𝑣_𝑠=1𝐶^𝐻_𝑠𝑅(𝑘)𝐷^𝐻_𝑠^𝑇+^𝑢_𝑙=1𝑃𝐴^𝐻_𝑙𝑅(𝑘)𝐵^𝐻_𝑙𝑄+^𝑣_𝑠=1𝑃𝐶^𝐻_𝑠𝑅(𝑘)𝐷^𝐻_𝑠^𝑇𝑄𝑇(𝑘)(𝑘)+‖𝑅(𝑘=𝑆)‖²‖𝑅(𝑘−1)‖²𝑇(𝑘−1). (18)
(5) Go to Step (3).

The following Lemmas show some properties of Algorithm 6.

Lemma 7. The sequences {𝑅(𝑖)},  {𝑆(𝑖)}, and {𝑇(𝑖)} generated by Algorithm 6 satisfy Retr𝑅^𝐻𝑅(𝑖𝑅𝑗+1)=Retr^𝐻𝑗𝑅(𝑖−𝑅𝑗₂)𝑇𝑗²Retr𝑇^𝐻𝑗𝑆(𝑖),Retr𝑇^𝐻𝑇(𝑖=𝑅𝑗+1)𝑗+1²𝑅𝑗²Retr𝑇^𝐻𝑗𝑇(𝑖+‖𝑇(𝑖))‖²‖𝑅(𝑖)‖²Retr𝑅^𝐻𝑅(𝑖𝑅𝑗+1)−Retr^𝐻𝑗+1𝑅(𝑖+1),𝑓𝑜𝑟𝑖,𝑗=1,2,…. (19)(20)

Proof. One has the following: Retr𝑅^𝐻𝑅(𝑖𝑅𝑗−𝑅𝑗𝑗+1)=Retr²𝑇𝑗²×^𝑢_𝑙=1𝐴_𝑙𝑇𝑗𝐵_𝑙+^𝑣_𝑠=1𝐶_𝑠𝑇^𝑇𝑗𝐷_𝑠^𝐻𝑅(𝑖)=Retr𝑅^𝐻𝑗𝑅(𝑖−𝑅𝑗)²𝑇𝑗²×Retr^𝑢_𝑙=1𝐵^𝐻_𝑙𝑇^𝐻𝑗𝐴^𝐻_𝑙𝑅(𝑖)+^𝑣_𝑠=1𝐷^𝐻_𝑠𝑇𝑗𝐶^𝐻_𝑠𝑅(𝑖𝑅)=Retr^𝐻𝑗𝑅(𝑖−𝑅𝑗)²𝑇𝑗²×Retr𝑇^𝐻𝑗^𝑢_𝑙=1𝐴^𝐻_𝑙𝑅(𝑖)𝐵^𝐻_𝑙+^𝑣_𝑠=1𝐶^𝐻_𝑠𝑅(𝑖)𝐷^𝐻_𝑠^𝑇=Retr𝑅^𝐻𝑗𝑅(𝑖−𝑅𝑗)²𝑇𝑗²×Retr𝑇^𝐻𝑗×^𝑢_𝑙=1𝐴^𝐻_𝑙𝑅(𝑖)𝐵^𝐻_𝑙+^𝑣_𝑠=1𝐶^𝐻_𝑠𝑅(𝑖)𝐷^𝐻_𝑠^𝑇+^𝑢_𝑙=1𝑃𝐴^𝐻_𝑙𝑅(𝑖)𝐵^𝐻_𝑙𝑄+^𝑣_𝑠=1𝑃𝐶^𝐻_𝑠𝑅(𝑖)𝐷^𝐻_𝑠^𝑇𝑄×2⁻¹=Retr𝑅^𝐻𝑗𝑅(𝑖−𝑅𝑗₂)𝑇𝑗²Retr𝑇^𝐻𝑗𝑆(𝑖). (21)
Therefore, the conclusion (19) holds. Consider Retr𝑇^𝐻𝑇(𝑖𝑆𝑗+1)=Retr^𝐻+𝑅𝑗+1𝑗+1²𝑅𝑗²𝑇^𝐻𝑗𝑇(𝑖=𝑅)𝑗+1²𝑅𝑗²Retr𝑇^𝐻𝑗𝑇(𝑖1)+Retr2^𝑢_𝑙=1𝐴^𝐻_𝑙𝑅𝑗+1𝐵^𝐻_𝑙+^𝑣_𝑠=1𝐶^𝐻_𝑠𝑅𝑗+1𝐷^𝐻_𝑠^𝑇+^𝑢_𝑙=1𝑃𝐴^𝐻_𝑙𝑅𝑗+1𝐵^𝐻_𝑙𝑄+^𝑣_𝑠=1𝑃𝐶^𝐻_𝑠𝑅𝑗+1𝐷^𝐻_𝑠^𝑇𝑄^𝐻(𝑖)=𝑅×𝑇𝑗+1²𝑅𝑗²Retr𝑇^𝐻𝑗𝑇(𝑖𝑇)+Retr^𝐻(𝑖)^𝑢_𝑙=1𝐴^𝐻_𝑙𝑅𝑗+1𝐵^𝐻_𝑙+^𝑣_𝑠=1𝐶^𝐻_𝑠𝑅𝑗+1𝐷^𝐻_𝑠^𝑇=𝑅𝑗+1²𝑅𝑗²Retr𝑇^𝐻𝑗𝑇(𝑖)+Retr^𝑢_𝑙=1𝑅^𝐻𝑗+1𝐴_𝑙𝑇(𝑖)𝐵_𝑙+^𝑣_𝑠=1𝑇(𝑖)𝐶^𝐻_𝑠𝑅𝑗+1𝐷^𝐻_𝑠=𝑅𝑗+1²𝑅𝑗²Retr𝑇^𝐻𝑗𝑇(𝑖𝑅)+Retr^𝐻𝑗+1^𝑢_𝑙=1𝐴_𝑙𝑇(𝑖)𝐵_𝑙+^𝑣_𝑠=1𝐶_𝑠𝑇^𝑇(𝑖)𝐷_𝑠=𝑅𝑗+1²𝑅𝑗²Retr𝑇^𝐻𝑗𝑇(𝑖+‖𝑇(𝑖))‖²‖𝑅(𝑖)‖²Retr𝑅^𝐻(𝑅(𝑖)(=𝑅𝑗+1−𝑅𝑖+1))𝑗+1²𝑅𝑗²Retr𝑇^𝐻𝑗𝑇(𝑖+‖𝑇(𝑖))‖²‖𝑅(𝑖)‖²Retr𝑅^𝐻𝑅(𝑖𝑅𝑗+1)−Retr^𝐻𝑗+1𝑅(𝑖+1). (22)
Therefore, the conclusion (20) holds.

Lemma 8. Assume that the sequences {𝑅(𝑖)} and {𝑇(𝑖)} are generated by Algorithm 6, and then ⟨𝑅(𝑖),𝑅(𝑗)⟩=0 and ⟨𝑇(𝑖),𝑇(𝑗)⟩=0 for 𝑖,𝑗=1,2,…,𝑖≠𝑗.

Proof. Since ⟨𝑅(𝑖),𝑅(𝑗)⟩=⟨𝑅(𝑗),𝑅(𝑖)⟩ and ⟨𝑇(𝑖),𝑇(𝑗)⟩=⟨𝑇(𝑗),𝑇(𝑖)⟩ for 𝑖,𝑗=1,2,…, we only need to prove that ⟨𝑅(𝑖),𝑅(𝑗)⟩=0 and ⟨𝑇(𝑖),𝑇(𝑗)⟩=0 for 1≤𝑖<𝑗.
Now we prove this conclusion by induction.
First, we prove ⟨𝑅(𝑖),𝑅(𝑖+1)⟩=0,⟨𝑇(𝑖),𝑇(𝑖+1)⟩=0for𝑖=1,2,…. (23) When 𝑖=1, from Lemma 7 we have ⟨𝑅(1),𝑅(2)⟩=Retr𝑅^𝐻(2)𝑅(1𝑅)=Retr^𝐻(1)𝑅(1−‖𝑅(1))‖²‖𝑇(1)‖²Retr𝑇^𝐻(1)𝑆(1=‖𝑅(1))‖²−‖𝑅(1)‖²‖𝑇(1)‖²‖𝑇(1)‖²=0. (24) From Lemma 7 and conclusion (24), we have ⟨𝑇(1),𝑇(2)⟩=Retr𝑇^𝐻(2)𝑇(1=‖𝑅(2))‖²‖𝑅(1)‖²Retr𝑇^𝐻(1)𝑇(1+‖𝑇(1))‖²‖𝑅(1)‖²Retr𝑅^𝐻(2)𝑅(1𝑅)−Retr^𝐻(2)𝑅(2=‖𝑅(2))‖²‖𝑅(1)‖²‖𝑇(1)‖²−‖𝑇(1)‖²‖𝑅(1)‖²‖𝑅(2)‖²=0. (25) Now assume that conclusion (23) holds for 1≤𝑖≤𝑡−1; from Lemma 7, we have ⟨𝑅(𝑡)(𝑅,𝑅𝑡+1)⟩=Retr^𝐻()𝑅(𝑡𝑅𝑡+1)=Retr^𝐻(𝑡)𝑅(𝑡−‖𝑅(𝑡))‖²‖𝑇(𝑡)‖²Retr𝑇^𝐻(𝑡)𝑆(𝑡𝑅)=Retr^𝐻(𝑡)𝑅(𝑡−‖𝑅(𝑡))‖²‖𝑇(𝑡)‖²×Retr𝑇^𝐻(𝑡)𝑇(𝑡)−‖𝑅(𝑡)‖²‖𝑅(𝑡−1)‖²𝑇(𝑡−1)=0. (26) From Lemma 7 and conclusion (26), it can also be obtained that ⟨𝑇(𝑡)(𝑇,𝑇𝑡+1)⟩=Retr^𝐻()𝑇(𝑡=‖𝑅(𝑡+1)𝑡+1)‖²‖𝑅(𝑡)‖²Retr𝑇^𝐻(𝑡)𝑇(𝑡+‖𝑇(𝑡))‖²‖𝑅(𝑡)‖²Retr𝑅^𝐻()𝑅(𝑡𝑅𝑡+1)−Retr^𝐻()𝑅(=‖𝑅(𝑡+1𝑡+1)𝑡+1)‖²‖𝑅(𝑡)‖²‖𝑇(𝑡)‖²−‖𝑇(𝑡)‖²‖𝑅(𝑡)‖²‖𝑅(𝑡+1)‖²=0. (27) Therefore, by the principle of induction, conclusion (23) holds.
Next, assume that ⟨𝑅(𝑖),𝑅(𝑖+𝑟)⟩=0 and ⟨𝑇(𝑖),𝑇(𝑖+𝑟)⟩=0 for 𝑖≥1 and 𝑟≥1. We will prove that ⟨𝑅(𝑖)(⟨𝑇(𝑖)(,𝑅𝑖+𝑟+1)⟩=0,,𝑇𝑖+𝑟+1)⟩=0. (28) In order to prove conclusion (28), we first prove that ⟨𝑅(1)(⟨𝑇(1)(,𝑅𝑟+2)⟩=0,,𝑇𝑟+2)⟩=0. (29) From Lemma 7 we have ⟨𝑅(1)(𝑅,𝑅𝑟+2)⟩=Retr^𝐻()𝑅(1𝑅𝑟+2)=Retr^𝐻()𝑅(1−‖𝑅(𝑟+1)𝑟+1)‖²‖𝑇(𝑟+1)‖²Retr𝑇^𝐻()𝑆(1𝑅𝑟+1)=Retr^𝐻()𝑅(1−‖𝑅(𝑟+1)𝑟+1)‖²‖𝑇(𝑟+1)‖²Retr𝑇^𝐻(𝑟+1)𝑇(1)=0. (30) From Lemma 7 and conclusion (30), we have ⟨𝑇(1)(𝑇,𝑇𝑟+2)⟩=Retr^𝐻()𝑇(1=‖𝑅(𝑟+2)𝑟+2)‖²‖𝑅(𝑟+1)‖²Retr𝑇^𝐻()𝑇(1+‖𝑇(1𝑟+1))‖²‖𝑅(1)‖²Retr𝑅^𝐻()𝑅(1𝑅𝑟+2)−Retr^𝐻(𝑟+2)𝑅(2)=0. (31) Also we have ⟨𝑅(𝑖)(𝑅,𝑅𝑖+𝑟+1)⟩=Retr^𝐻()𝑅(𝑖𝑅𝑖+𝑟+1)=Retr^𝐻()𝑅(𝑖−‖𝑅(𝑖+𝑟)𝑖+𝑟)‖²‖𝑇(𝑖+𝑟)‖²Retr𝑇^𝐻()𝑆(𝑖‖𝑅(𝑖+𝑟)=−𝑖+𝑟)‖²‖𝑇(𝑖+𝑟)‖²Retr𝑇^𝐻()×𝑇(𝑖)−‖𝑅(𝑖𝑖+𝑟)‖²‖𝑅(𝑖−1)‖²()=‖𝑅(×𝑇𝑖−1𝑖+𝑟)‖²‖𝑅(𝑖)‖²‖𝑇(𝑖+𝑟)‖²‖𝑅(𝑖−1)‖²×Retr𝑇^𝐻()𝑇(,⟨𝑇(𝑖)(𝑇𝑖+𝑟𝑖−1),𝑇𝑖+𝑟+1)⟩=Retr^𝐻()𝑇(𝑖=‖𝑅(𝑖+𝑟+1)𝑖+𝑟+1)‖²‖𝑅(𝑖+𝑟)‖²Retr𝑇^𝐻()𝑇(𝑖+‖𝑇(𝑖𝑖+𝑟))‖²‖𝑅(𝑖)‖²Retr𝑅^𝐻()𝑅(𝑖𝑅𝑖+𝑟+1)−Retr^𝐻()(=‖‖𝑖+𝑟+1×𝑅𝑖+1)𝑇(𝑖)²‖𝑅(𝑖)‖²Retr𝑅^𝐻()𝑅(𝑖=‖‖𝑖+𝑟+1)𝑇(𝑖)²‖𝑅(𝑖+𝑟)‖²‖𝑅(𝑖)‖²‖𝑅(𝑖)‖²‖𝑇(𝑖+𝑟)‖²‖𝑅(𝑖−1)‖²×Retr𝑇^𝐻(𝑖+𝑟)𝑇(𝑖−1). (32) Repeating the previous process (32), we have ⟨𝑅(𝑖),𝑅(𝑖+𝑟+1)⟩=⋯=𝛼Retr𝑇^𝐻()𝑇(1,⟨𝑇(𝑖)(𝑇𝑟+2),𝑇𝑖+𝑟+1)⟩=⋯=𝛽Retr^𝐻(𝑟+2)𝑇(1). (33) Combining these two relations with (30) and (31), it implies that (28) holds. So, by the principle of induction, we know that Lemma 8 holds.

Lemma 9. When Problem 1 is consistent, let 𝑋∈ℍ_𝑟^𝑛×𝑛(𝑃,𝑄) be its solution; then, for any generalized (𝑃,𝑄)-reflexive initial matrix 𝑋(1)∈ℍ_𝑟^𝑛×𝑛(𝑃,𝑄), the sequences {𝑅(𝑖)}, {𝑇(𝑖)} and {𝑋(𝑖)} generated by Algorithm 6 satisfy 𝑇(𝑖),𝑋−𝑋(𝑖)=‖𝑅(𝑖)‖²,𝑖=1,2,…. (34)

Proof. When 𝑖=1, it follows from Algorithm 6 that 𝑇(1),(1(1)𝑋−𝑋)=Retr𝑋−𝑋^𝐻𝑇(1(1))=Retr𝑋−𝑋^𝐻×^𝑢_𝑙=1𝐴^𝐻_𝑙𝑅(1)𝐵^𝐻_𝑙+^𝑣_𝑠=1𝐶^𝐻_𝑠𝑅(1)𝐷^𝐻_𝑠^𝑇+^𝑢_𝑙=1𝑃𝐴^𝐻_𝑙𝑅(1)𝐵^𝐻_𝑙𝑄+^𝑣_𝑠=1𝑃𝐶^𝐻_𝑠𝑅(1)𝐷^𝐻_𝑠^𝑇𝑄×2⁻¹=Retr𝑋−𝑋(1)^𝐻𝑢_𝑙=1𝐴^𝐻_𝑙𝑅(1)𝐵^𝐻_𝑙+^𝑣_𝑠=1𝐶^𝐻_𝑠𝑅(1)𝐷^𝐻_𝑠^𝑇=Retr𝑅^𝐻(1)^𝑢_𝑙=1𝐴_𝑙𝑋−𝑋(1)𝐵_𝑙+^𝑣_𝑠=1𝐶_𝑠𝑋^𝑇−𝑋^𝑇(1)𝐷_𝑠=Retr𝑅^𝐻(1)𝑅(1=‖𝑅(1))‖². (35) This implies that (34) holds for 𝑖=1.
Assume that (34) holds for 𝑖=𝑡, and then, when 𝑖=𝑡+1, 𝑇(),(()𝑡+1𝑋−𝑋𝑡+1)=Retr𝑋−𝑋𝑡+1^𝐻𝑇(𝑡+1)=Retr𝑋−𝑋(𝑡+1)^𝐻×12^𝑢_𝑙=1𝐴^𝐻_𝑙𝑅(𝑡+1)𝐵^𝐻_𝑙+^𝑣_𝑠=1𝐶^𝐻_𝑠𝑅(𝑡+1)𝐷^𝐻_𝑠^𝑇+^𝑢_𝑙=1𝑃𝐴^𝐻_𝑙𝑅(𝑡+1)𝐵^𝐻_𝑙𝑄+^𝑣_𝑠=1𝑃𝐶^𝐻_𝑠𝑅(𝑡+1)𝐷^𝐻_𝑠^𝑇𝑄+‖𝑅(𝑡+1)‖²‖𝑅(𝑡)‖²𝑇(𝑡)=Retr𝑋−𝑋(𝑡+1)^𝐻×^𝑢_𝑙=1𝐴^𝐻_𝑙𝑅(𝑡+1)𝐵^𝐻_𝑙+^𝑣_𝑠=1𝐶^𝐻_𝑠𝑅(𝑡+1)𝐷^𝐻_𝑠^𝑇+‖𝑅(𝑡+1)‖²‖𝑅(𝑡)‖²Retr𝑋−𝑋(𝑡+1)^𝐻𝑇(𝑡𝑅)=Retr^𝐻()×𝑡+1_𝑢𝑙=1𝐴_𝑙𝑋−𝑋(𝑡+1)𝐵_𝑙+^𝑣_𝑠=1𝐶_𝑠𝑋^𝑇−𝑋^𝑇(𝑡+1)𝐷_𝑠+‖𝑅(𝑡+1)‖²‖𝑅(𝑡)‖²×Retr𝑋−𝑋(𝑡)^𝐻𝑇(𝑡𝑋()(𝑡)−Retr(𝑡+1−𝑋))^𝐻𝑇(𝑡=‖𝑅()𝑡+1)‖²+‖𝑅(𝑡+1)‖²‖𝑅(𝑡)‖²‖𝑅(𝑡)‖²−‖𝑅(𝑡)‖²‖𝑇(𝑡)‖²Retr𝑇^𝐻(𝑡)𝑇(𝑡=‖𝑅()𝑡+1)‖². (36) Therefore, Lemma 9 holds by the principle of induction.

Remark 10. Lemma 9 implies that if there exists a positive integer 𝑖 such that 𝑅(𝑖)≠0 and 𝑇(𝑖)=0, then Problem 1 is not consistent. Therefore, the solvability of Problem 1 can be determined by Algorithm 6 in the absence of round-off errors.

From Lemmas 8 and 9, we have the following theorem.

Theorem 11. Assume that Problem 1 is consistent, and then using Algorithm 6 for any generalized (𝑃,𝑄)-reflexive initial matrix 𝑋(1)∈ℍ_𝑟^𝑛×𝑛(𝑃,𝑄), a solution of Problem 1 can be obtained within finite iteration steps in the absence of round-off errors.

Proof. If 𝑅(𝑖)≠0,𝑖=1,2,…,4𝑝𝑞, from Lemma 9 we can obtain 𝑇(𝑖)≠0,𝑖=1,2,…,4𝑝𝑞. Thus 𝑅(4𝑝𝑞+1) and 𝑇(4𝑝𝑞+1) can be computed by Algorithm 6. According to Lemma 8, we have ⟨𝑅(𝑖),𝑅(𝑗)⟩=0, 𝑖,𝑗=1,2,…,4𝑝𝑞. Considering that the inner product space (ℍ^𝑝×𝑞,ℝ,⟨⋅,⋅⟩) is 4𝑝𝑞-dimensional, we know that the set of 𝑅(1),𝑅(2),…,𝑅(4𝑝𝑞) is an orthogonal basis of the inner product space (ℍ^𝑝×𝑞,ℝ,⟨⋅,⋅⟩). Also from Lemma 8, we have ⟨𝑅(𝑖),𝑅(4𝑝𝑞+1)⟩=0 for 𝑖=1,2,…,4𝑝𝑞, which implies that 𝑅(4𝑝𝑞+1)=0; that is, 𝑋(4𝑝𝑞+1) is a solution of Problem 1.

4. The Solution of Problem 2

In this section, we will prove that the sequence {𝑋(𝑘)} generated by Algorithm 6 converges to the least Frobenius norm generalized (𝑃,𝑄)-reflexive solution of the quaternion matrix equation ∑^𝑢_𝑙=1𝐴_𝑙𝑋𝐵_𝑙+∑^𝑣_𝑠=1𝐶_𝑠𝑋^𝑇𝐷_𝑠=𝐹 when an appropriate initial iterative matrix is chosen. Then, we solve Problem 2 by finding the least Frobenius norm generalized (𝑃,𝑄)-reflexive solution of a new constructed quaternion matrix equation.

Lemma 12 (see [51]). Assume that the consistent system of linear equations 𝑀𝑦=𝑏 has a solution 𝑦₀∈𝑅(𝑀^𝑇); then, 𝑦₀ is the unique least Frobenius norm solution of the system of linear equations.

Lemma 13. If Problem 1 is consistent, then the following system of real linear equations is consistent: ^𝑢∑_𝑙=1𝜙^𝑇𝐵_𝑙⊗𝜙𝐴_𝑙−∑^𝑣_𝑠=1𝜙^𝑇𝐷_𝑠⊗𝜙𝐶_𝑠𝑃(4𝑛,4𝑛)+2∑^𝑣_𝑠=1𝜙^𝑇𝐷_𝑠⊗𝜙𝐶_𝑠diag(vec(𝑊))𝑃(4𝑛,4𝑛_𝑢)_𝑙∑₌₁𝜙^𝑇𝐵_𝑙𝜙(𝑄)⊗𝜙𝐴_𝑙𝜙(𝑃)−∑^𝑣_𝑠=1𝜙^𝑇𝐷_𝑠𝜙(𝑃)⊗𝜙𝐶_𝑠𝜙(𝑄)𝑃(4𝑛,4𝑛)+2∑^𝑣_𝑠=1𝜙^𝑇𝐷_𝑠⊗𝜙𝐶_𝑠((𝑊𝜙(𝑃)(𝑄)𝑃()𝜙(𝐹)𝜙(𝐹).×diagvec))⊗𝜙4𝑛,4𝑛𝑌=vecvec (37)
Furthermore, if the solution sets of Problem 1 and (37) are denoted by 𝑆_𝑋 and 𝑆_𝑌, respectively, then, vec𝜙𝑆_𝑋⊆𝑆_𝑌. (38)

Proof. If Problem 1 is consistent, let 𝑋 be a solution of Problem 1, then we have ∑^𝑢_𝑙=1𝐴_𝑙𝑋𝐵_𝑙+∑^𝑣_𝑠=1𝐶_𝑠𝑋^𝑇𝐷_𝑠=𝐹 and 𝑃𝑋𝑄=𝑋, which implies that 𝑋 is a solution of quaternion matrix equations ^𝑢_𝑙=1𝐴_𝑙𝑋𝐵_𝑙+^𝑣_𝑠=1𝐶_𝑠𝑋^𝑇𝐷_𝑠=𝐹,_𝑢𝑙=1𝐴_𝑙𝑃𝑋𝑄𝐵_𝑙+^𝑣_𝑠=1𝐶_𝑠(𝑃𝑋𝑄)^𝑇𝐷_𝑠=𝐹. (39)
By (8), (9), and (12), we can derive that the quaternion matrix equations (39) is equivalent to the following real matrix equations: ^𝑢_𝑙=1𝜙𝐴_𝑙𝜙(𝑋)𝜙𝐵_𝑙−^𝑣_𝑠=1𝜙𝐶_𝑠𝜙^𝑇(𝑋)𝜙𝐷_𝑠+2^𝑣_𝑠=1𝜙𝐶_𝑠𝜙^𝑇(𝑋)⊙𝑊𝜙𝐷_𝑠=𝜙_𝑢𝑙=1(𝐹),𝜙𝐴_𝑙𝜙(𝑃)𝜙(𝑋)𝜙(𝑄)𝜙𝐵_𝑙−^𝑣_𝑠=1𝜙𝐶_𝑠𝜙(𝑄)𝜙^𝑇(𝑋)𝜙(𝑃)𝜙𝐷_𝑠+2^𝑣_𝑠=1𝜙𝐶_𝑠𝜙(𝑄)𝜙^𝑇(𝑋)𝜙(𝑃)⊙𝑊𝜙𝐷_𝑠=𝜙(𝐹). (40)
By the operation properties of vec(⋅),  ⊗, ⊙, and Lemma 4, it is not difficult to verify that matrix equations (40) is equivalent to^𝑢∑_𝑙=1𝜙^𝑇𝐵_𝑙⊗𝜙𝐴_𝑙−∑^𝑣_𝑠=1𝜙^𝑇𝐷_𝑠⊗𝜙𝐶_𝑠𝑃(4𝑛,4𝑛)+2∑^𝑣_𝑠=1𝜙^𝑇𝐷_𝑠⊗𝜙𝐶_𝑠diag(vec(𝑊))𝑃(4𝑛,4𝑛_𝑢)_𝑙∑₌₁𝜙^𝑇𝐵_𝑙𝜙(𝑄)⊗𝜙𝐴_𝑙𝜙(𝑃)−∑^𝑣_𝑠=1𝜙^𝑇𝐷_𝑠𝜙(𝑃)⊗𝜙𝐶_𝑠𝜙(𝑄)𝑃(4𝑛,4𝑛)+2∑^𝑣_𝑠=1𝜙^𝑇𝐷_𝑠⊗𝜙𝐶_𝑠((𝑊𝜙(𝑃)(𝑄)𝑃()𝜙(𝑋)=𝜙(𝐹)𝜙(𝐹),×diagvec))⊗𝜙4𝑛,4𝑛×vecvecvec (41)which implies that vec(𝜙(𝑋)) is a solution of (37). So system of real linear equations (37) is consistent.
From the previous procedure, the proof of (38) is trivial.

Theorem 14. If 𝑋^∘ is a solution of Problem 1 and 𝑋^∘ can be expressed as the following form: 𝑋^∘=^𝑢_𝑙=1𝐴^𝐻_𝑙𝐺𝐵^𝐻_𝑙+^𝑣_𝑠=1𝐶^𝐻_𝑠𝐺𝐷^𝐻_𝑠^𝑇+^𝑢_𝑙=1𝑃𝐴^𝐻_𝑙𝐺𝐵^𝐻_𝑙𝑄+^𝑣_𝑠=1𝑃𝐶^𝐻_𝑠𝐺𝐷^𝐻_𝑠^𝑇𝑄,𝐺∈ℍ^𝑝×𝑞 (42) then, 𝑋^∘ is the least Frobenius norm solution of Problem 1.

Proof. By (8), (9), (10), and (12) and Lemmas 4 and 5, we havevec𝜙𝑋^∘=vec^𝑢_𝑙=1𝜙^𝑇𝐴_𝑙𝜙(𝐺)𝜙^𝑇𝐵_𝑙+2^𝑣_𝑠=1𝜙𝐷_𝑠𝜙^𝑇(𝐺)𝜙𝐶_𝑠−⊙𝑊_𝑣𝑠=1𝜙𝐷_𝑠𝜙^𝑇(𝐺)𝜙𝐶_𝑠+^𝑢_𝑙=1𝜙(𝑃)𝜙^𝑇𝐴_𝑙𝜙(𝐺)𝜙^𝑇𝐵_𝑙𝜙(𝑄)+2^𝑣_𝑠=1𝜙(𝑃)𝜙𝐷_𝑠𝜙^𝑇(𝐺)𝜙𝐶_𝑠⊙𝑊𝜙(𝑄)−^𝑣_𝑠=1𝜙(𝑃)𝜙𝐷_𝑠𝜙^𝑇(𝐺)𝜙𝐶_𝑠𝜙(𝑄)=^𝑢_𝑙=1𝜙𝐵_𝑙⊗𝜙^𝑇𝐴_𝑙+2^𝑣_𝑠=1diag(vec(𝑊))𝜙^𝑇𝐶_𝑠⊗𝜙𝐷_𝑠𝑃−4𝑝,4𝑞_𝑣𝑠=1𝜙^𝑇𝐶_𝑠⊗𝜙𝐷_𝑠𝑃4𝑝,4𝑞_𝑢𝑙=1,𝜙(𝑄)𝜙𝐵_𝑙⊗𝜙(𝑃)𝜙^𝑇𝐴_𝑙+2^𝑣_𝑠=1𝜙(𝑄)⊗𝜙(𝑃)diag(vec(𝑊))𝜙^𝑇𝐶_𝑠⊗𝜙𝐷_𝑠𝑃−4𝑝,4𝑞_𝑣𝑠=1𝜙(𝑄)𝜙^𝑇𝐶_𝑠⊗𝜙(𝑃)𝜙𝐷_𝑠𝑃𝜙(𝐺𝜙(𝐺=4𝑝,4𝑞vec)vec)_𝑢𝑙∑₌₁𝜙^𝑇𝐵_𝑙⊗𝜙𝐴_𝑙−∑^𝑣_𝑠=1𝜙^𝑇𝐷_𝑠⊗𝜙𝐶_𝑠𝑃(4𝑛,4𝑛)+2∑^𝑣_𝑠=1𝜙^𝑇𝐷_𝑠⊗𝜙𝐶_𝑠diag(vec(𝑊))𝑃(4𝑛,4𝑛_𝑢)_𝑙∑₌₁𝜙^𝑇𝐵_𝑙𝜙(𝑄)⊗𝜙𝐴_𝑙𝜙(𝑃)−∑^𝑣_𝑠=1𝜙^𝑇𝐷_𝑠𝜙(𝑃)⊗𝜙𝐶_𝑠𝜙(𝑄)𝑃(4𝑛,4𝑛)+2∑^𝑣_𝑠=1𝜙^𝑇𝐷_𝑠⊗𝜙𝐶_𝑠diagvec(𝑊)𝜙(𝑃)⊗𝜙(𝑄)𝑃(4𝑛,4𝑛)^𝑇×𝜙(𝐺)𝜙(𝐺)vecvec∈𝑅^𝑢_𝑙∑₌₁𝜙^𝑇𝐵_𝑙⊗𝜙𝐴_𝑙−∑^𝑣_𝑠=1𝜙^𝑇𝐷_𝑠⊗𝜙𝐶_𝑠𝑃(4𝑛,4𝑛)+2∑^𝑣_𝑠=1𝜙^𝑇𝐷_𝑠⊗𝜙𝐶_𝑠diag(vec(𝑊))𝑃(4𝑛,4𝑛_𝑢)_𝑙∑₌₁𝜙^𝑇𝐵_𝑙𝜙(𝑄)⊗𝜙𝐴_𝑙𝜙(𝑃)−∑^𝑣_𝑠=1𝜙^𝑇𝐷_𝑠𝜙(𝑃)⊗𝜙𝐶_𝑠𝜙(𝑄)𝑃(4𝑛,4𝑛)+2∑^𝑣_𝑠=1𝜙^𝑇𝐷_𝑠⊗𝜙𝐶_𝑠diagvec(𝑊)𝜙(𝑃)⊗𝜙(𝑄)𝑃(4𝑛,4𝑛)^𝑇. (43)By Lemma 12, vec(𝜙(𝑋^∘)) is the least Frobenius norm solution of (37).
Noting (11), we derive from Lemma 13 that 𝑋^∘ is the least Frobenius norm solution of Problem 1.

Remark 15. From Algorithm 6, we can derive that if we choose 𝑋(1)=∑^𝑢_𝑙=1𝐴^𝐻_𝑙𝐺𝐵^𝐻_𝑙+∑^𝑣_𝑠=1(𝐶^𝐻_𝑠𝐺𝐷^𝐻_𝑠)^𝑇+∑^𝑢_𝑙=1𝑃𝐴^𝐻_𝑙𝐺𝐵^𝐻_𝑙𝑄+∑^𝑣_𝑠=1𝑃(𝐶^𝐻_𝑠𝐺𝐷^𝐻_𝑠)^𝑇𝑄,  𝐺∈ℍ^𝑝×𝑞, as the initial iteration matrix, then all 𝑋(𝑘) generated by Algorithm 6 have a form of 𝑋(𝑘)=^𝑢_𝑙=1𝐴^𝐻_𝑙𝐺_𝑘𝐵^𝐻_𝑙+^𝑣_𝑠=1𝐶^𝐻_𝑠𝐺_𝑘𝐷^𝐻_𝑠^𝑇+^𝑢_𝑙=1𝑃𝐴^𝐻_𝑙𝐺_𝑘𝐵^𝐻_𝑙𝑄+^𝑣_𝑠=1𝑃𝐶^𝐻_𝑠𝐺_𝑘𝐷^𝐻_𝑠^𝑇𝑄,𝐺_𝑘∈ℍ^𝑝×𝑞. (44)

Theorem 16. When Problem 1 is consistent, if we choose the following matrix as the initial iteration matrix 𝑋(1)=^𝑢_𝑙=1𝐴^𝐻_𝑙𝐺𝐵^𝐻_𝑙+^𝑣_𝑠=1𝐶^𝐻_𝑠𝐺𝐷^𝐻_𝑠^𝑇+^𝑢_𝑙=1𝑃𝐴^𝐻_𝑙𝐺𝐵^𝐻_𝑙𝑄+^𝑣_𝑠=1𝑃𝐶^𝐻_𝑠𝐺𝐷^𝐻_𝑠^𝑇𝑄, (45) where 𝐺∈ℍ^𝑝×𝑞 is arbitrary, or especially, 𝑋(1)=0, then the solution 𝑋^∗, generated by Algorithm 6, is exactly the least Frobenius norm solution of Problem 1.

From conclusion (44) and Theorem 14, the proof of Theorem 16 is trivial.

Now we consider Problem 2. When Problem 1 is consistent, the solution set of Problem 1  𝑆_𝑋 is not empty. For a given generalized (𝑃,𝑄)-reflexive matrix 𝑋₀∈ℍ_𝑟^𝑛×𝑛(𝑃,𝑄), let 𝑋̇=𝑋−𝑋₀,  𝐹̇=𝐹−∑^𝑢_𝑙=1𝐴_𝑙𝑋₀𝐵_𝑙−∑^𝑣_𝑠=1𝐶_𝑠𝑋^𝑇₀𝐷_𝑠; then the least Frobenius norm generalized (𝑃,𝑄)-reflexive solution of the following quaternion matrix equation is exactly the solution of min_{𝑋∈𝑆𝑋}‖𝑋−𝑋₀‖^𝑢_𝑙=1𝐴_𝑙𝑋̇𝐵_𝑙+^𝑣_𝑠=1𝐶_𝑠𝑋̇^𝑇𝐷_𝑠=𝐹̇. (46) According to Theorem 16, if we choose the initial iteration matrix 𝑋̇(1)=∑^𝑢_𝑙=1𝐴^𝐻_𝑙𝐺𝐵^𝐻_𝑙+∑^𝑣_𝑠=1(𝐶^𝐻_𝑠𝐺𝐷^𝐻_𝑠)^𝑇+∑^𝑢_𝑙=1𝑃𝐴^𝐻_𝑙𝐺𝐵^𝐻_𝑙𝑄+∑^𝑣_𝑠=1𝑃(𝐶^𝐻_𝑠𝐺𝐷^𝐻_𝑠)^𝑇𝑄, where 𝐺∈ℍ^𝑝×𝑞 is arbitrary, or especially, choosing𝑋̇(1)=0, the least Frobenius norm generalized (𝑃,𝑄)-reflexive solution 𝑋̇^∗ of (46) can be obtained. Then the solution of Problem 2 is 𝑋=𝑋̇^∗+𝑋₀.

5. Examples

In this section, we give two examples to illustrate the performance of the proposed algorithm.

Example 17. Consider the quaternion matrix equation 𝐴₁𝑋𝐵₁+𝐶₁𝑋^𝑇𝐷₁+𝐴₂𝑋𝐵₂+𝐶₂𝑋^𝑇𝐷₂=𝐹, (47) with the matrices𝐴₁=,𝐴2+𝑖+2𝑗1+2𝑖+2𝑗+𝑘3𝑗2𝑖+𝑗+2𝑘2𝑖+2𝑗+𝑘−1+3𝑘1+2𝑖+4𝑗+2𝑘4+𝑖+2𝑗+4𝑘₂=,𝐵𝑖+𝑗+𝑘𝑖1−2𝑖+𝑗+2𝑘2+4𝑖+2𝑘2𝑖+2𝑗+𝑘1+3𝑖+𝑗3+4𝑖+𝑗−2𝑘2+2𝑖+2𝑗+𝑘₁=1+5𝑖+2𝑗+𝑘2+3𝑖+3𝑗+2𝑘6𝑖+4𝑗−𝑘2+5𝑗2+2𝑖+𝑗−𝑘4+7𝑖+2𝑗+2𝑘3+𝑗−𝑘1+2𝑖−2𝑗+3𝑘,𝐵₂=,𝐶−𝑖+4𝑗+𝑘−3+2𝑖+3𝑗+2𝑘2+4𝑖−𝑘−1+5𝑖+2𝑗−2𝑘1+6𝑖+4𝑗+5𝑘2−3𝑖+𝑗+𝑘−𝑖+6𝑗+3𝑘1+7𝑖−2𝑗−4𝑘₁=,𝐶𝑖+2𝑗+𝑘1+𝑖−3𝑗+2𝑘1+5𝑖+𝑘2𝑖+4𝑗−4𝑘1−4𝑖+2𝑗+𝑘5+5𝑖−𝑗+𝑘−4+2𝑖+3𝑗+2𝑘1+5𝑖+2𝑗−4𝑘₂=,𝐷1+𝑖+3𝑗3−𝑖+3𝑗+𝑘1+𝑖+5𝑗+3𝑘2+4𝑖+2𝑗5+3𝑖+𝑗+4𝑘2+𝑖+2𝑗+3𝑘3+4𝑖+3𝑗+2𝑘−2+𝑖−3𝑗+𝑘₁=2+6𝑖+6𝑗+𝑘1+3𝑖+𝑗−2𝑘−4𝑖+3𝑗+8𝑘−2−2𝑖−4𝑗+5𝑘5+3𝑖−2𝑗+4𝑘1+2𝑖+𝑗+2𝑘3−𝑖+2𝑗−2𝑘3+𝑖−𝑗+3𝑘,𝐷₂=,.𝑖−5𝑗+6𝑘−2−8𝑖−4𝑗−3𝑘3+4𝑖+7𝑗+2𝑘5𝑖+2𝑗5+7𝑖−2𝑗−𝑘−2+9𝑖+7𝑗+𝑘1+3𝑖+4𝑗6+2𝑗+7𝑘𝐹=6+𝑖+2𝑗+𝑘−8−7𝑖−3𝑗+2𝑘𝑖−6𝑗−4𝑘3−2𝑖+6𝑘 (48) We apply Algorithm 6 to find the generalized (𝑃,𝑄)-reflexive solution of (47), where 𝑃=,0.2800.96𝑘000.280−0.96𝑗−0.96𝑘0−0.28000.96𝑗0−0.28𝑄=0.2800.96𝑘00−100−0.96𝑘0−0.280000−1 (49) are two generalized reflection matrices. Choosing an initial matrix 𝑋(1)=100000.3600.48𝑗00100−0.48𝑗00.64∈ℍ_𝑟^4×4(𝑃,𝑄), (50) we obtain that the sequence {𝑋(𝑘)} after 21 steps is𝑋^∗()==𝑋21_{0.51783+0.066965𝑖+0.095973𝑗−0.025639𝑘−0.088601+0.063970𝑖+0.075293𝑗+0.15420𝑘−0.13818−0.050488𝑖−0.073437𝑗+0.11425𝑘0.12924−0.077969𝑖−0.12809𝑗+0.12008𝑘0.026384+0.045766𝑖−0.22964𝑗+0.066304𝑘−0.20560−0.10039𝑖+0.085294𝑗−0.11813𝑘0.12175+0.14209𝑖−0.11665𝑗+0.20690𝑘−0.17079−0.16010𝑖−0.17233𝑗−0.10396𝑘−0.026384+0.045766𝑖−0.22964𝑗−0.066304𝑘−0.034186+0.089643𝑖−0.0029795𝑗+0.11576𝑘−0.14209−0.12175𝑖+0.20690𝑗−0.11665𝑘−0.080702+0.12324𝑖+0.20981𝑗+0.11867𝑘0.55651+0.066993𝑖−0.069276𝑗−0.041030𝑘−0.15435+0.0039727𝑖+0.11952𝑗−0.045581𝑘0.070202−0.070130𝑖−0.031364𝑗+0.0024134𝑘0.27974−0.15822𝑖+0.10760𝑗+0.16431𝑘}, (51)with ‖𝑅(21)‖=2.047×10⁻¹³. The obtained results are presented in Figure 1, where 𝑟(𝑘)=‖𝑅(𝑘)‖.

Figure 1 

The curve for the Frobenius norm of the residuals from Example 17.

Example 18. Let𝑋₀=0.64−0.72𝑖−0.75𝑘0.48𝑘−0.5625𝑖−0.96𝑘0.64+0.36𝑖00.48𝑗+0.48𝑘0−0.48𝑘1−0.96𝑗0.361.28−0.75𝑗0.48𝑗+0.48𝑘00.64+0.36𝑖0∈ℍ_𝑟^4×4(𝑃,𝑄). (52)Now we consider Problem 2, that is, finding the optimal approximate generalized (𝑃,𝑄)-reflexive solution of the quaternion matrix equation (47) to 𝑋₀. Applying Algorithm 6 to (46) by choosing the initial matrix 𝑋̇(1)=0, we obtain that the sequence {𝑋(𝑘)} after 22 steps is𝑋̇^∗=𝑋̇()=,22−0.60797+0.14851𝑖−0.031273𝑗+0.032158𝑘0.093923+0.086087𝑖+0.24350𝑗+0.44225𝑘−0.32942−0.34127𝑖−0.091587𝑗+0.36047𝑘−0.092590−0.15547𝑖−0.19614𝑗+0.067121𝑘−0.087200−0.13538𝑖+0.090255𝑗+0.25660𝑘−0.58966−0.32467𝑖+0.11478𝑗+0.12523𝑘0.020667+0.14412𝑖−0.17405𝑗+0.058232𝑘−0.26152−0.089494𝑖+0.12345𝑗−0.20729𝑘0.087200−0.13538𝑖+0.090255𝑗−0.25660𝑘−0.021779+0.31853𝑖−0.55293𝑗+0.58717𝑘−0.14412−0.020667𝑖+0.058232𝑗−0.17405𝑘0.090225−0.057613𝑖−0.17618𝑗−0.066567𝑘−0.45828−0.20116𝑖−0.047696𝑗+0.083025𝑘−0.78290+0.73724𝑖+0.42471𝑗−0.029038𝑘−0.30730−0.22789𝑖−0.27640𝑗+0.079531𝑘−0.23490+0.088756𝑖−0.12030𝑗−0.076818𝑘 (53)with ‖𝑅̇(22)‖=2.1855×10⁻¹⁴. The obtained results are presented in Figure 2, where 𝑟(𝑘)=‖𝑅̇(𝑘)‖.
So we can obtain the solution of Problem 2; that is,𝑋=𝑋̇^∗+𝑋₀=.0.032031+0.14851𝑖−0.031273𝑗+0.032158𝑘0.093923−0.63391𝑖+0.24350𝑗−0.30775𝑘0.31058+0.018728𝑖−0.091587𝑗+0.36047𝑘−0.092590−0.15547𝑖−0.19614𝑗+0.067121𝑘−0.087200−0.13538𝑖+0.090255𝑗−0.22340𝑘0.41034−0.32467𝑖−0.84522𝑗+0.12523𝑘0.020667+0.14412𝑖+0.30595𝑗+0.53823𝑘−0.26152−0.089494𝑖+0.12345𝑗−0.20729𝑘0.087200−0.13538𝑖+0.090255𝑗+0.22340𝑘−0.021779−0.24397𝑖−0.55293𝑗−0.37283𝑘−0.14412−0.020667𝑖+0.53823𝑗+0.30595𝑘0.090225−0.057613𝑖−0.17618𝑗−0.066567𝑘−0.098285−0.20116𝑖−0.047696𝑗+0.083025𝑘0.49710+0.73724𝑖−0.32529𝑗−0.029038𝑘0.33270+0.13211𝑖−0.27640𝑗+0.079531𝑘−0.23490+0.088756𝑖−0.12030𝑗−0.076818𝑘 (54)

Figure 2 

The curve for the Frobenius norm of the residuals from Example 18.

6. Conclusions

In this paper, an algorithm has been presented for solving the quaternion matrix equation ∑^𝑢_𝑙=1𝐴_𝑙𝑋𝐵_𝑙+∑^𝑣_𝑠=1𝐶_𝑠𝑋^𝑇𝐷_𝑠=𝐹 over generalized (𝑃,𝑄)-reflexive matrices. It has been proven that when the quaternion matrix equation ∑^𝑢_𝑙=1𝐴_𝑙𝑋𝐵_𝑙+∑^𝑣_𝑠=1𝐶_𝑠𝑋^𝑇𝐷_𝑠=𝐹 is consistent over generalized (𝑃,𝑄)-reflexive matrices, for any generalized (𝑃,𝑄)-reflexive initial iterative matrix, the sequence {𝑋(𝑘)} generated by Algorithm 6 will converge to a generalized (𝑃,𝑄)-reflexive solution within finite iteration steps in the absence of round-off errors. We have also proven that by choosing a suitable initial iterative matrix, the sequence {𝑋(𝑘)} will converge to the least Frobenius norm generalized (𝑃,𝑄)-reflexive solution of the quaternion matrix equation ∑^𝑢_𝑙=1𝐴_𝑙𝑋𝐵_𝑙+∑^𝑣_𝑠=1𝐶_𝑠𝑋^𝑇𝐷_𝑠=𝐹. Then, by using Algorithm 6, we solved Problem 2. The numerical results have shown that the presented algorithm is quite effective.

Acknowledgments

This research was supported by the grants from the Innovation Program of Shanghai Municipal Education Commission (13ZZ080), the National Natural Science Foundation of China (11171205), and the Natural Science Foundation of Shanghai (11ZR1412500).