The Hermitian 𝑅-Conjugate Generalized Procrustes Problem

Abstract

We consider the Hermitian 𝑅-conjugate generalized Procrustes problem to find Hermitian 𝑅-conjugate matrix 𝑋 such that ∑^𝑝_𝑘=1‖𝐴_𝑘𝑋−𝐶_𝑘‖²+∑^𝑞_𝑙=1‖𝑋𝐵_𝑙−𝐷_𝑙‖² is minimum, where 𝐴_𝑘, 𝐶_𝑘, 𝐵_𝑙, and 𝐷_𝑙 (𝑘=1,2,…,𝑝, 𝑙=1,…,𝑞) are given complex matrices, and 𝑝 and 𝑞 are positive integers. The expression of the solution to Hermitian 𝑅-conjugate generalized Procrustes problem is derived. And the optimal approximation solution in the solution set for Hermitian 𝑅-conjugate generalized Procrustes problem to a given matrix is also obtained. Furthermore, we establish necessary and sufficient conditions for the existence and the formula for Hermitian 𝑅-conjugate solution to the linear system of complex matrix equations 𝐴₁𝑋=𝐶₁, 𝐴₂𝑋=𝐶₂,…,𝐴_𝑝𝑋=𝐶_𝑝, 𝑋𝐵₁=𝐷₁,…,𝑋𝐵_𝑞=𝐷_𝑞 (𝑝 and 𝑞 are positive integers). The representation of the corresponding optimal approximation problem is presented. Finally, an algorithm for solving two problems above is proposed, and the numerical examples show its feasibility.

1. Introduction

Throughout, let 𝒞^𝑚×𝑛 denote the set of all complex 𝑚×𝑛 matrices, ℛ^𝑚×𝑛 the set of all real 𝑚×𝑛 matrices, and ℛ_𝑟^𝑚×𝑛 the set of all matrices in ℛ^𝑚×𝑛 with rank 𝑟. The symbols 𝐼, 𝐴, 𝐴^𝑇, 𝐴^∗, 𝐴^†, and ‖𝐴‖, respectively, stand for the identity matrix with the appropriate size, the conjugate, the transpose, the conjugate transpose, the Moore-Penrose inverse, and the Frobenius norm of 𝐴∈𝒞^𝑚×𝑛. For 𝐴=(𝑎_𝑖𝑗), 𝐵=(𝑏_𝑖𝑗)∈𝒞^𝑚×𝑛; 𝐴∗𝐵=(𝑎_𝑖𝑗𝑏_𝑖𝑗)∈𝒞^𝑚×𝑛 represents the Hadamard product of 𝐴 and 𝐵.

A linear model of image restoration is a matrix-vector equation is 𝑦=𝐾𝑥+𝑛, (1) where 𝑦 represents the observed image, 𝑥 the original true image, 𝑛 additive noise, and 𝐾 a blurring matrix. Image restoration is to minimize blur in an observed image, namely, recover a optimal approximation of 𝑥 by given 𝑦 and 𝐾, and get some statistical information of 𝑛. The process of the image restoration for the model (1) can be described as min_𝑥‖𝑥‖²subjectto𝐾𝑥−𝑦²=𝜀, (2) where 𝜀 is a small positive parameter. It is known that 𝑥=𝐾^†𝑦 is the least squares solution of (1) with minimal norm. However, for 𝜀=0, the solution 𝑥_𝜀 to (2), that is, 𝑛=0 in (1) is not feasible. We know if 𝜀→0, then the solution 𝑥_𝜀 converges to 𝑥. In order to obtain a solution 𝑥_𝜀 sufficiently near to 𝑥, we usually take small 𝜀 such that 0<𝜀≪𝛿, where the error norm 𝛿∶=‖𝑛‖ is given.

Now we consider the generalized problem of the process of the image restoration.

Problem 1. Given 𝒮⊆𝒞^𝑛×𝑛, 𝑚_𝑘, ℎ_𝑙, 𝑝, 𝑞 being positive integers, 𝐴_𝑘,𝐶_𝑘∈𝒞^{𝑚𝑘×𝑛}, 𝐵_𝑙,𝐷_𝑙∈𝒞^{𝑛×ℎ𝑙}, 𝐸∈𝒞^𝑛×𝑛, and 𝑘=1,…,𝑝, 𝑙=1,…,𝑞. Let 𝓒=𝑋∈𝒮∣^𝑝_𝑘=1𝐴_𝑘𝑋−𝐶_𝑘²+^𝑞_𝑙=1𝑋𝐵_𝑙−𝐷_𝑙²=min. (3) Find 𝑋∈𝓒 such that 𝑋−𝐸=min_𝑋∈𝓒‖𝑋−𝐸‖. (4)
The constraint Procrustes problem associated with several kinds of sets 𝒮, that is, 𝑝=1 and 𝑞=0 in (3) has been extensively studied, such as the orthogonal Procrustes problem with 𝒮 being the set of orthogonal matrices [1], the symmetric Procrustes problem [2], (𝑀,𝑁)-symmetric Procrustes problem [3], Hermitian, Hermitian 𝑅-symmetric and Hermitian 𝑅-skew-symmetric Procrustes problems [4], the Procrustes problems with 𝒮 constrained to the cone of symmetric positive semidefinite and symmetric elementwise matrices [5], and the generalized Procrustes analysis [6]. The optimal approximation problem (4) is initially proposed in the processes of testing or revising given data. A preliminary estimate 𝐸 of the unknown matrix 𝑋 in 𝓒 can be obtained from experimental observation values and the information of statistical distribution.

We characterize the case 𝐴_𝑘𝑋=𝐶_𝑘, 𝑋𝐵_𝑙=𝐷_𝑙, 𝑘=1,…,𝑝, 𝑙=1,…,𝑞 in Problem 1 and describe it as follows.

Problem 2. Given 𝒮⊆𝒞^𝑛×𝑛, 𝑚₁,…,𝑚_𝑝, ℎ₁,…,ℎ_𝑞, 𝑝 and 𝑞 being positive integers, 𝐴₁,𝐶₁∈𝒞^𝑚1×𝑛, 𝐴₂,𝐶₂∈𝒞^𝑚2×𝑛,…,𝐴_𝑝,𝐶_𝑝∈𝒞^{𝑚𝑝×𝑛}, 𝐵₁,𝐷₁∈𝒞^𝑛×ℎ1,…,𝐵_𝑞,𝐷_𝑞∈𝒞^{𝑛×ℎ𝑞}, 𝐹∈𝒞^𝑛×𝑛. Let 𝓛=𝑋∈𝒮∣𝐴₁𝑋=𝐶₁,𝐴₂𝑋=𝐶₂,…,𝐴_𝑝𝑋=𝐶_𝑝,𝑋𝐵₁=𝐷₁,…,𝑋𝐵_𝑞=𝐷_𝑞. (5) When 𝓛 is nonempty, find 𝑋∈𝓛 such that 𝑋−𝐹=min_𝑋∈𝓛‖𝑋−𝐹‖. (6)

For important results to solve Problem 2 with different sets 𝒮, we refer to [7–14].

Motivated by the work mentioned above, in this paper we mainly discuss the above two problems associated with 𝑆 being the set of Hermitian 𝑅-conjugate matrices.

Recall that an 𝑛×𝑛 complex matrix 𝐾 is 𝑅-conjugate if 𝐾=𝑅𝐾𝑅, where 𝑅∈ℛ^𝑛×𝑛 is a nontrivial involution, that is, 𝑅²=𝐼, 𝑅≠±𝐼, which was defined in [15]. A matrix 𝐾∈𝒞^𝑛×𝑛 is Hermitian 𝑅-conjugate if 𝐾^∗=𝐾 and 𝐾=𝑅𝐾𝑅, where 𝑅^𝑇=𝑅⁻¹=𝑅≠±𝐼. The Hermitian 𝑅-conjugate matrix is very useful in scientific computation and digital signal and image processing, its special case, for example, Hermitian Toeplitz matrix, have been studied by several authors, see [14, 16–21]. We denote the set of all 𝑛×𝑛 Hermitian 𝑅-conjugate matrices by 𝐻𝑅_𝑐𝒞^𝑛×𝑛. Let 𝑅_𝑐𝒞^𝑛×𝑛, 𝑆ℛ^𝑛×𝑛, and 𝐴𝑆ℛ^𝑛×𝑛 denote the set of all 𝑛×𝑛 complex 𝑅-conjugate matrices, real symmetric matrices, real skew-symmetric matrices, respectively.

This paper is organized as follows. In Section 2, we give some preliminary lemmas. In Section 3, we derive the expression of the unique solution to the Problem 1 with 𝒮=𝐻𝑅_𝑐𝒞^𝑛×𝑛. In Section 4, we establish the solvability conditions for existence and an expression of the solution for Problem 2 with 𝒮=𝐻𝑅_𝑐𝒞^𝑛×𝑛. In Section 5, we give examples to illustrate the results obtained in this paper.

2. Preliminaries

In order to study Problems 1 and 2 with 𝒮=𝐻𝑅_𝑐𝒞^𝑛×𝑛, we first give some preliminary lemmas in this section.

For a nontrivial symmetric involutory matrix 𝑅∈ℛ^𝑛×𝑛, there exist positive integers 𝑟 and 𝑠 such that 𝑟+𝑠=𝑛 and an 𝑛×𝑛 orthogonal matrix [𝑃,𝑄] such that 𝐼𝑅=𝑃,𝑄^𝑟00−𝐼_𝑠𝑃^𝑇𝑄^𝑇, (7) where 𝑃∈ℛ^𝑛×𝑟 and 𝑄∈ℛ^𝑛×𝑠. The columns of 𝑃(𝑄) form an orthogonal basis for the eigenspace of 𝑅 associated with the eigenvalue 1(−1).

Throughout this paper, we always suppose the nontrivial symmetric involutory matrix 𝑅 is fixed which is given by (7).

Lemma 3 (see Theorem 2.1 in [14]). A matrix 𝐾∈𝑅_𝑐𝒞^𝑛×𝑛 if and only if there exists 𝐻₁∈ℛ^𝑛×𝑛 such that 𝐾=𝑈𝐻₁𝑈^∗, and 𝐾∈𝐻𝑅_𝑐𝒞^𝑛×𝑛 if and only if there exists 𝐻₂∈𝑆ℛ^𝑛×𝑛 such that 𝐾=𝑈𝐻₂𝑈^∗, where 𝑈=𝑃,−𝑖𝑄, (8) with 𝑃 and 𝑄 being the same as (7).

Lemma 4. For any matrix 𝐴∈𝒞^𝑛×𝑛, 𝐴=𝐴₁⊕𝐴₂⊕𝑖𝐴₃, where 𝐵=𝐴+𝑅𝐴𝑅2,𝐴₁=𝐵+𝐵^∗2,𝐴₂=𝐵−𝐵^∗2,𝐴₃=𝐴−𝑅𝐴𝑅,2𝑖 (9) and ⊕ denotes the direct sum.

Proof. For 𝐵∈𝒞^𝑛×𝑛, it is obvious that 𝐵=𝐴₁⊕𝐴₂, where 𝐴₁ are 𝐴₂ are defined in (9). Hence, we just need to prove 𝐴=𝐵⊕𝑖𝐴₃.
For any matrix 𝐴∈𝒞^𝑛×𝑛, it is obvious that 𝐴=𝐵+𝑖𝐴₃, where 𝐵 and 𝐴₃ are defined in (9).
We prove the uniqueness of 𝐴=𝐵+𝑖𝐴₃. Note that 𝑅𝐵𝑅=𝐵,𝑅𝐴₃𝑅=𝐴₃, (10) that is, 𝐵,𝐴₃∈𝑅_𝑐𝒞^𝑛×𝑛. If there exist 𝐷 and 𝐶₃ satisfying 𝑅𝐷𝑅=𝐷,𝑅𝐶₃𝑅=𝐶₃ (11) such that 𝐴=𝐷+𝑖𝐶₃, then 𝐵−𝐷=𝑖𝐶₃−𝐴₃. (12) Multiplying 𝑅 on the left and right side and then taking the conjugate for (12), it yields 𝐵−𝐷=𝑖𝐴₃−𝐶₃, (13) implying 𝐵=𝐷 and 𝐴₃=𝐶₃.
So 𝐴=𝐴₁⊕𝐴₂⊕𝑖𝐴₃ holds, where 𝐴₁, 𝐴₂, and 𝐴₃ are defined in (9).

Lemma 5. Let the symmetric involution 𝑅∈ℛ^𝑛×𝑛 be given in (7) and let 𝑈 be defined as (8). Then(i)a matrix 𝐴∈𝒞^𝑚×𝑛 satisfies 𝐴𝑅=𝐴 if and only if 𝐴𝑈∈𝑅^𝑚×𝑛,(ii)a matrix 𝐵∈𝒞^𝑛×𝑙 satisfies 𝑅𝐵=𝐵 if and only if  𝑈^∗𝐵∈𝑅^𝑛×𝑙.

Proof. (i) It yields from (7) and (8) that 𝐼𝑅=𝑃,−𝑖𝑄^𝑟00−𝐼_𝑠𝑃^𝑇𝑖𝑄^𝑇𝐼=𝑈^𝑟00−𝐼_𝑠𝑈^∗. (14) By (14) 𝐼𝐴=𝐴𝑅=𝐴𝑈^𝑟00−𝐼_𝑠𝑈^∗, (15) that is, 𝐴=𝐼𝐴𝑈^𝑟00−𝐼_𝑠𝑈^∗=𝐴𝑈𝑈^∗, (16) implying 𝐴𝑈=𝐴𝑈, that is, 𝐴𝑈∈𝑅^𝑚×𝑛.
Conversely, if 𝐴𝑈∈𝑅^𝑚×𝑛, according to the proof of the necessity, we can get 𝐴=𝐴𝑅.
The proof of (ii) can be analogously completed according to the proof of (i).

3. The Solution to Problem 1 with 𝒮=𝐻𝑅_𝑐𝒞^𝑛×𝑛

We, in this section, give the explicit expression of the solution to Problem 1 with 𝒮=𝐻𝑅_𝑐𝒞^𝑛×𝑛. In the following, we refer to the 𝒮=𝐻𝑅_𝑐𝒞^𝑛×𝑛 in 𝓒.

According to Lemma 3, if 𝑋∈𝐻𝑅_𝑐𝒞^𝑛×𝑛, then 𝑋=𝑈𝑌𝑈^∗, (17) where 𝑈 is defined as (8) and 𝑌∈𝑆ℛ^𝑛×𝑛. Let 𝐴_𝑘=𝐴_𝑘1+𝑖𝐴_𝑘2∈𝒞^{𝑚𝑘×𝑛},𝐴_𝑘1=𝐴_𝑘+𝐴_𝑘𝑅2,𝐴_𝑘2=𝐴_𝑘−𝐴_𝑘𝑅,2𝑖𝐵_𝑙=𝐵_𝑙1+𝑖𝐵_𝑙2∈𝒞^{𝑛×ℎ𝑙},𝐵_𝑙1=𝐵_𝑙+𝑅𝐵_𝑙2,𝐵_𝑙2=𝐵_𝑙−𝑅𝐵_𝑙,2𝑖𝐶_𝑘𝑈=𝐶_𝑘1+𝑖𝐶_𝑘2,𝐶_𝑘1,𝐶_𝑘2∈ℛ^{𝑚𝑘×𝑛},𝑈^∗𝐷_𝑙=𝐷_𝑙1+𝑖𝐷_𝑙2,𝐷_𝑙1,𝐷_𝑙2∈ℛ^{𝑛×ℎ𝑙}. (18)(19)(20) By Lemma 5, it is easy to verify 𝐴_𝑘1𝑈,𝐴_𝑘2𝑈∈ℛ^{𝑚𝑘×𝑛}, 𝑈^∗𝐵_𝑙1,𝑈^∗𝐵_𝑙2∈ℛ^{𝑛×ℎ𝑙}. We obtain 𝐴_𝑘𝑋−𝐶_𝑘=𝐴_𝑘1+𝑖𝐴_𝑘2𝑈𝑌𝑈^∗−𝐶_𝑘=𝐴_𝑘1𝑈𝑌−𝐶_𝑘1+𝑖𝐴_𝑘2𝑈𝑌−𝐶_𝑘2𝑈^∗,𝑋𝐵_𝑙−𝐷_𝑙=𝑈𝑌𝑈^∗𝐵_𝑙1+𝑖𝐵_𝑙2−𝐷_𝑙=𝑈𝑌𝑈^∗𝐵_𝑙1−𝐷_𝑙1+𝑖𝑌𝑈^∗𝐵_𝑙2−𝐷_𝑙2. (21)(22) It follows from the unitary invariance of Frobenius norm, (21), (22), and 𝑌∈𝑆ℛ^𝑛×𝑛 that ^𝑝_𝑘=1𝐴_𝑘𝑋−𝐶_𝑘²+^𝑞_𝑙=1𝑋𝐵_𝑙−𝐷_𝑙²=^𝑝_𝑘=1𝐴_𝑘1𝑈𝑌−𝐶_𝑘1²+𝐴_𝑘2𝑈𝑌−𝐶_𝑘2²+^𝑞_𝑙=1𝑌𝑈^∗𝐵_𝑙1−𝐷_𝑙1²+𝑌𝑈^∗𝐵_𝑙2−𝐷_𝑙2²=^𝑝_𝑘=1𝐴_𝑘1𝑈𝐴_𝑘2𝑈𝐶𝑌−^𝑘1𝐶_𝑘2²+^𝑞_𝑙=1𝑌𝑈^∗𝐵_𝑙1,𝑈^∗𝐵_𝑙2−𝐷_𝑙1,𝐷_𝑙2²=^𝑝_𝑘=1𝐴_𝑘1𝑈𝐴_𝑘2𝑈𝐶𝑌−^𝑘1𝐶_𝑘2²+^𝑞_𝑙=1(𝑈^∗𝐵_𝑙1)^𝑇(𝑈^∗𝐵_𝑙2)^𝑇𝐷𝑌−^𝑇𝑙1𝐷^𝑇_𝑙2². (23) Suppose 𝑆=𝐴₁₁𝑈^𝑇,𝐴₁₂𝑈^𝑇,…,𝐴_𝑝1𝑈^𝑇,𝐴_𝑝2𝑈^𝑇,𝑈^∗𝐵₁₁,𝑈^∗𝐵₁₂,…,𝑈^∗𝐵_𝑞1,𝑈^∗𝐵_𝑞2^𝑇,𝐺=𝐶^𝑇₁₁,𝐶^𝑇₁₂,…,𝐶^𝑇_𝑝1,𝐶^𝑇_𝑝2,𝐷₁₁,𝐷₁₂,…,𝐷_𝑞1,𝐷_𝑞2^𝑇, (24)(25) then ^𝑝_𝑘=1𝐴_𝑘𝑋−𝐶_𝑘²+^𝑞_𝑙=1𝑋𝐵_𝑙−𝐷_𝑙²=‖𝑆𝑌−𝐺‖². (26)

We first give the following lemma which can be obtained by contrast with Lemma 2.1 in [22].

Lemma 6. Given 𝑆,𝐺∈ℛ^{(𝑚1+⋯+𝑚𝑝+ℎ1+⋯+ℎ𝑞)×𝑛}; therefore, the singular value decomposition (SVD) of 𝑆∈ℛ^{(𝑚1+⋯+𝑚𝑝+ℎ1+⋯+ℎ𝑞}_𝑟^)×𝑛₁ can be described as Σ𝑆=𝑊^𝑟10𝑉00^𝑇=𝑊₁𝐷_𝑟1𝑉^𝑇₁, (27) where 𝑊=𝑊₁,𝑊₂,𝑉=𝑉₁,𝑉₂, (28) Let 𝑊₁∈ℛ^{(𝑚1+⋯+𝑚𝑝+ℎ1+⋯+ℎ𝑞)×𝑟1}, 𝑊₂∈ℛ^{(𝑚1+⋯+𝑚𝑝+ℎ1+⋯+ℎ𝑞)×(𝑚1+⋯+𝑚𝑝+ℎ1+⋯+ℎ𝑞−𝑟1)}, 𝑉₁∈ℛ^𝑛×𝑟1, 𝑉₂∈ℛ^{𝑛×(𝑛−𝑟1)}, and Σ_𝑟1=diag(𝑑₁,…,𝑑_𝑟1) with 𝑑_𝑖>0, 𝑖=1,…,𝑟₁; Φ=(1/(𝑑²_𝑖+𝑑²_𝑗))_𝑟1×𝑟1. Then min_{𝑌∈𝑆ℛ}^𝑛×𝑛‖𝑆𝑌−𝐺‖ is consistent if and only if Σ𝑌=𝑉Φ∗_𝑟1𝑊^𝑇₁𝐺𝑉₁+𝑉^𝑇₁𝐺^𝑇𝑊₁Σ_𝑟1Σ_𝑟⁻¹₁𝑊^𝑇₁𝐺𝑉₂𝑉^𝑇₂𝐺^𝑇𝑊₁Σ_𝑟⁻¹₁𝐿𝑉^𝑇, (29) where 𝐿∈𝑆ℛ^{(𝑛−𝑟1)×(𝑛−𝑟1)} is arbitrary.

Theorem 7. Given 𝐴_𝑘,𝐶_𝑘∈𝒞^{𝑚𝑘×𝑛}, 𝐵_𝑙,𝐷_𝑙∈𝒞^{𝑛×ℎ𝑙}, and positive integers 𝑚_𝑘, ℎ_𝑙, 𝑝, and 𝑞, where 𝑘=1,…,𝑝, 𝑙=1,…,𝑞, the notations 𝐴_𝑘1, 𝐴_𝑘2, 𝐵_𝑙1, 𝐵_𝑙2, 𝐶_𝑘1, 𝐶_𝑘2, 𝐷_𝑙1, 𝐷_𝑙2, 𝑆, 𝐺 are defined as (18), (19), (20), (24), and (25), respectively. Let the SVD of 𝑆∈ℛ^{(𝑚1+⋯+𝑚𝑝+ℎ1+⋯+ℎ𝑞}_𝑟^)×𝑛₁ be of the form (27) with (28). Then 𝓒=𝐿∈𝑆ℛ^{(𝑛−𝑟1)×(𝑛−𝑟1)}×Σ∣𝑈𝑉Φ∗_𝑟1𝑊^𝑇₁𝐺𝑉₁+𝑉^𝑇₁𝐺^𝑇𝑊₁Σ_𝑟1Σ_𝑟⁻¹₁𝑊^𝑇₁𝐺𝑉₂𝑉^𝑇₂𝐺^𝑇𝑊₁Σ_𝑟⁻¹₁𝐿×𝑉^𝑇𝑈^∗. (30)

Proof. It yields from (26) that min_{𝑋∈𝐻𝑅𝑐𝒞𝑛×𝑛}^𝑝_𝑘=1𝐴_𝑘𝑋−𝐶_𝑘²+^𝑞_𝑙=1𝑋𝐵_𝑙−𝐷_𝑙²=min_{𝑌∈𝑆ℛ𝑛×𝑛}‖𝑆𝑌−𝐺‖². (31) By Lemma 6, min_{𝑌∈𝑆ℛ}^𝑛×𝑛‖𝑆𝑌−𝐺‖ is consistent if and only if 𝑌 has the expression of (29). Taking (29) into (17), we obtain the solution set 𝓒 is (30).

Theorem 8. Given 𝐸∈𝒞^𝑛×𝑛, the equation (4) is consistent if and only if ×Σ𝑋=𝑈𝑉Φ∗_𝑟1𝑊^𝑇₁𝐺𝑉₁+𝑉^𝑇₁𝐺^𝑇𝑊₁Σ_𝑟1Σ_𝑟1⁻¹𝑊^𝑇₁𝐺𝑉₂𝑉^𝑇₂𝐺^𝑇𝑊₁Σ_𝑟1⁻¹𝑉^𝑇₂𝑈^∗𝐸+𝐸^∗𝑈𝑉₂2×𝑉^𝑇𝑈^∗. (32)

Proof. Obviously, 𝓒 is a closed convex set. Hence, there exists the unique element 𝑋∈𝓛 such that (4) holds. By applying Theorem 7 and the unitary invariance of Frobenius norm, for 𝑋∈𝓒, we get ‖𝑋−𝐸‖²=Σ𝑈𝑉Φ∗_𝑟1𝑊^𝑇₁𝐺𝑉₁+𝑉^𝑇₁𝐺^𝑇𝑊₁Σ_𝑟1Σ_𝑟⁻¹₁𝑊^𝑇₁𝐺𝑉₂𝑉^𝑇₂𝐺^𝑇𝑊₁Σ_𝑟⁻¹₁𝐿×𝑉^𝑇𝑈^∗−𝐸²=Φ∗Σ_𝑟1𝑊^𝑇₁𝐺𝑉₁+𝑉^𝑇₁𝐺^𝑇𝑊₁Σ_𝑟1Σ_𝑟⁻¹₁𝑊^𝑇₁𝐺𝑉₂𝑉^𝑇₂𝐺^𝑇𝑊₁Σ_𝑟⁻¹₁𝐿−𝑉^𝑇𝑈^∗𝐸𝑈𝑉²=Φ∗Σ_𝑟1𝑊^𝑇₁𝐺𝑉₁+𝑉^𝑇₁𝐺^𝑇𝑊₁Σ_𝑟1−𝑉^𝑇₁𝑈^∗𝐸𝑈𝑉₁²+Σ_𝑟⁻¹₁𝑊^𝑇₁𝐺𝑉₂−𝑉^𝑇₁𝑈^∗𝐸𝑈𝑉₂²+𝑉^𝑇₂𝐺^𝑇𝑊₁Σ_𝑟⁻¹₁−𝑉^𝑇₂𝑈^∗𝐸𝑈𝑉₁²+𝐿−𝑉^𝑇₂𝑈^∗𝐸𝑈𝑉₂². (33) Then min_𝑋∈𝓒‖𝑋−𝐸‖ is equivalent to min_{𝐿∈𝑆ℛ11)(𝑛−𝑟)×(𝑛−𝑟}𝐿−𝑉^𝑇₂𝑈^∗𝐸𝑈𝑉₂. (34) (34) holds if and only if 𝑉𝐿=^𝑇2𝑈^∗𝐸+𝐸^∗𝑈𝑉₂2. (35) Substituting (35) into 𝑋∈𝓒, we get 𝑋 is (32).

Corollary 9. ‖𝑋_min‖=min_𝑋∈𝓒‖𝑋‖ if and only if 𝑋_min×Σ=𝑈𝑉Φ∗_𝑟1𝑊^𝑇₁𝐺𝑉₁+𝑉^𝑇₁𝐺^𝑇𝑊₁Σ_𝑟1Σ_𝑟⁻¹₁𝑊^𝑇₁𝐺𝑉₂𝑉^𝑇₂𝐺^𝑇𝑊₁Σ_𝑟⁻¹₁0×𝑉^𝑇𝑈^∗. (36)

Remark 10. when 𝑝=1 and 𝑞=1 in Theorem 8, we can derive a result of Theorem 4.1 in [14].

4. The Solution to Problem 2 with 𝒮=𝐻𝑅_𝑐𝒞^𝑛×𝑛

We refer to 𝒮=𝐻𝑅_𝑐𝒞^𝑛×𝑛 in 𝓛 in the following text. In this section, we establish necessary and sufficient conditions for the existence and the expression of 𝓛. When 𝓛 is nonempty, we present the expression of the unique solution to (6).

It follows from (21) that the system 𝐴₁𝑋=𝐶₁, 𝐴₂𝑋=𝐶₂,…,𝐴_𝑝𝑋=𝐶_𝑝, 𝑋𝐵₁=𝐷₁,…,𝑋𝐵_𝑞=𝐷_𝑞 with unknown 𝑋∈𝐻𝑅_𝑐𝒞^𝑛×𝑛 is consistent if and only if there exists 𝑌∈𝑆ℛ^𝑛×𝑛 such that 𝑆𝑌=𝐺. We first give the following lemma.

Lemma 11 (see Theorem 1 in [7]). Given 𝑆,𝐺∈𝑅^{(𝑚1+⋯+𝑚𝑝+ℎ1+⋯+ℎ𝑞)×𝑛}. Let the SVD of 𝑆∈𝑅^{(𝑚1+⋯+𝑚𝑝+ℎ1+⋯+ℎ𝑞}_𝑟^)×𝑛₁ be (27) with (28). Then the matrix equation 𝑆𝑌=𝐺 has a symmetric solution if and only if 𝑆𝑆^†𝐺=𝐺,𝐺𝑆^𝑇=𝑆𝐺^𝑇, (37) and the symmetric solution can be expressed as 𝑌=𝑆^†𝐺+𝐼−𝑆^†𝑆𝑆^†𝐺^𝑇+𝑉₂𝑍𝑉^𝑇₂, (38) where 𝑍∈𝑆ℛ^{(𝑛−𝑟1)×(𝑛−𝑟1)} is arbitrary.

Theorem 12. Given 𝐴₁,𝐶₁∈𝒞^𝑚1×𝑛, 𝐴₂,𝐶₂∈𝒞^𝑚2×𝑛,…,𝐴_𝑝,𝐶_𝑝∈𝒞^{𝑚𝑝×𝑛}, 𝐵₁,𝐷₁∈𝒞^𝑛×ℎ1,…,𝐵_𝑞,𝐷_𝑞∈𝒞^{𝑛×ℎ𝑞}, and positive integers 𝑝 and 𝑞. The notations 𝐴_𝑘1, 𝐴_𝑘2, 𝐵_𝑙1, 𝐵_𝑙2, 𝐶_𝑘1, 𝐶_𝑘2, 𝐷_𝑙1, 𝐷_𝑙2, 𝑆, 𝐺 are defined as (18), (19), (20), (24), and (25), respectively. Let the SVD of 𝑆∈ℛ^{(𝑚1+⋯+𝑚𝑝+ℎ1+⋯+ℎ𝑞}_𝑟^)×𝑛₁ be of the form (27) with (28). Then the solution set 𝓛 is nonempty if and only if (37) holds, in which case, 𝓛=𝑍∈𝑆ℛ^{(𝑛−𝑟1)×(𝑛−𝑟1)}∣𝑈𝑆^†𝐺+𝐼−𝑆^†𝑆𝑆^†𝐺^𝑇+𝑉₂𝑍𝑉^𝑇₂𝑈^∗. (39)

Proof. If the solution set 𝓛 is nonempty, then there exists a matrix 𝑋∈𝐻𝑅_𝑐𝒞^𝑛×𝑛 such that 𝐴₁𝑋=𝐶₁, 𝐴₂𝑋=𝐶₂,…,𝐴_𝑝𝑋=𝐶_𝑝, 𝑋𝐵₁=𝐷₁,…,𝑋𝐵_𝑞=𝐷_𝑞. We know that 𝐴₁𝑋=𝐶₁, 𝐴₂𝑋=𝐶₂,…,𝐴_𝑝𝑋=𝐶_𝑝, 𝑋𝐵₁=𝐷₁,…,𝑋𝐵_𝑞=𝐷_𝑞 with 𝑋∈𝐻𝑅_𝑐𝒞^𝑛×𝑛 is consistent if and only if there exists 𝑌∈𝑆ℛ^𝑛×𝑛 such that 𝑆𝑌=𝐺. By Lemma 11, 𝑆𝑌=𝐺 is solvable for 𝑌∈𝑆ℛ^𝑛×𝑛 if and only if (37) holds, and the expression of the solution is (38). Insert (38) into (17), then we obtain 𝓛 is of the form (39).
Conversely, assume (37) holds, according to the proof of the necessity, 𝐴₁𝑋=𝐶₁, 𝐴₂𝑋=𝐶₂,…,𝐴_𝑝𝑋=𝐶_𝑝, 𝑋𝐵₁=𝐷₁,…,𝑋𝐵_𝑞=𝐷_𝑞 is solvable for 𝑋∈𝐻𝑅_𝑐𝒞^𝑛×𝑛. For 𝑍∈𝑆ℛ^{(𝑛−𝑟1)×(𝑛−𝑟1)}, by Lemma 3, 𝑈[𝑆^†𝐺+(𝐼−𝑆^†𝑆)(𝑆^†𝐺)^𝑇+𝑉₂𝑍𝑉^𝑇₂]𝑈^∗∈𝐻𝑅_𝑐𝒞^𝑛×𝑛.

Theorem 13. Given 𝐹∈𝒞^𝑛×𝑛 and 𝓛 is nonempty. Let 𝐹₁=(1/4)[𝐹+𝑅𝐹𝑅+(𝐹+𝑅𝐹𝑅)^∗]. Then (6) is consistent for 𝑋∈𝓛 if and only if 𝑋=𝑈𝑆^†𝐺+𝐼−𝑆^†𝑆𝑆^†𝐺^𝑇+𝑉₂𝑉^𝑇₂𝑈𝐹₁𝑈^∗𝑉₂𝑉^𝑇₂𝑈^∗. (40) In particular, if ‖𝑋_inf‖=min_𝑋∈𝓛‖𝑋‖, then 𝑋_inf=𝑈𝑆^†𝐺+𝐼−𝑆^†𝑆𝑆^†𝐺^𝑇𝑈^∗. (41)

Proof. By Lemma 4, 𝐹=𝐹₁⊕𝐹₂⊕𝑖𝐹₃, where 𝐹₁=14𝐹+𝑅𝐹𝑅+𝐹+𝑅𝐹𝑅^∗,𝐹₂=14𝐹+𝑅𝐹𝑅−𝐹+𝑅𝐹𝑅^∗,𝐹₃=𝐹−𝑅𝐹𝑅.2𝑖 (42) When 𝓛 is nonempty, for 𝑋∈𝓛, we get ‖𝑋−𝐹‖²=𝑈𝑆^†𝐺+𝐼−𝑆^†𝑆𝑆^†𝐺^𝑇+𝑉₂𝑍𝑉^𝑇₂𝑈^∗−𝐹₁+𝐹₂+𝑖𝐹₃²=𝑆^†𝐺+𝐼−𝑆^†𝑆𝑆^†𝐺^𝑇+𝑉₂𝑍𝑉^𝑇₂−𝑈^∗𝐹₁𝑈−𝑈^∗𝐹₂𝑈−𝑖𝑈^∗𝐹₃𝑈². (43) Since 𝐹₁∈𝐻𝑅_𝑐𝒞^𝑛×𝑛, 𝐹₂,𝐹₃∈𝑅_𝑐𝒞^𝑛×𝑛, by Lemma 3, we obtain 𝑈^∗𝐹₁𝑈∈𝑆ℛ^𝑛×𝑛, 𝑈^∗𝐹₂𝑈,𝑈^∗𝐹₃𝑈∈ℛ^𝑛×𝑛. Note that 𝑈^∗𝐹₂𝑈∈𝐴𝑆ℛ^𝑛×𝑛, then ‖𝑋−𝐹‖²=𝑈𝐹₁𝑈^∗−𝑆^†𝐺−𝐼−𝑆^†𝑆𝑆^†𝐺^𝑇−𝑉₂𝑍𝑉^𝑇₂²+𝑈𝐹₂𝑈^∗2+𝑈𝐹₃𝑈^∗2. (44) Hence, min_𝑋∈𝓛‖𝑋−𝐹‖ is equivalent to min_{𝑍∈𝑆ℛ11)(𝑛−𝑟)×(𝑛−𝑟}𝑈𝐹₁𝑈^∗−𝑆^†𝐺−𝐼−𝑆^†𝑆𝑆^†𝐺^𝑇−𝑉₂𝑍𝑉^𝑇₂. (45) For the orthogonal matrix 𝑉, we get 𝑈𝐹₁𝑈^∗−𝑆^†𝐺−𝐼−𝑆^†𝑆𝑆^†𝐺^𝑇−𝑉₂𝑍𝑉^𝑇₂=𝑉^𝑇𝑈𝐹₁𝑈^∗−𝑆^†𝐺−𝐼−𝑆^†𝑆𝑆^†𝐺^𝑇−𝑉₂𝑍𝑉^𝑇₂𝑉=𝑉^𝑇₁𝑈𝐹₁𝑈^∗−𝑆^†𝐺−𝐼−𝑆^†𝑆𝑆^†𝐺^𝑇𝑉₁+𝑉^𝑇₁𝑈𝐹₁𝑈^∗−𝑆^†𝐺−𝐼−𝑆^†𝑆𝑆^†𝐺^𝑇𝑉₂+𝑉^𝑇₂𝑈𝐹₁𝑈^∗−𝑆^†𝐺−𝐼−𝑆^†𝑆𝑆^†𝐺^𝑇𝑉₁+𝑉^𝑇₂𝑈𝐹₁𝑈^∗−𝑆^†𝐺−𝐼−𝑆^†𝑆𝑆^†𝐺^𝑇𝑉₂−𝑍. (46) Then min_{𝑍∈𝑆ℛ11}^{)(𝑛−𝑟)×(𝑛−𝑟}‖𝑈𝐹₁𝑈^∗−𝑆^†𝐺−(𝐼−𝑆^†𝑆)(𝑆^†𝐺)^𝑇−𝑉₂𝑍𝑉^𝑇₂‖ is solvable if and only if 𝑍=𝑉^𝑇₂𝑈𝐹₁𝑈^∗−𝑆^†𝐺−𝐼−𝑆^†𝑆𝑆^†𝐺^𝑇𝑉₂. (47) It follows from (27) that 𝑆^†=𝑉₁𝐷_𝑟⁻¹₁𝑊^𝑇₁. (48) By (48) and (47) we get 𝑍=𝑉^𝑇₂𝑈𝐹₁𝑈^∗𝑉₂. (49) Hence, from (39) and (49), we obtain 𝑋 which can be expressed as (40).

Remark 14. When 𝑝=1 and 𝑞=1 in Theorem 13, we can get a conclusion of Theorem 3.1 in [14].

5. Numerical Examples

We, in this section, propose an algorithm for finding the solution of Problems 1 and 2 with 𝒮=𝐻𝑅_𝑐𝒞^𝑛×𝑛 and give illustrative numerical examples.

Let the symmetric involutory matrix 𝑅=^00𝑉20𝐼₂⁰_𝑉200∈ℛ^6×6, where 𝑉₂=⁰¹₁₀. Applying the spectral decomposition of 𝑅, we obtain the orthogonal matrix [𝑃,𝑄] satisfying (7) and then by (8), we get 1𝑈=√2√00100−𝑖0001𝑖00√200000.200000001−𝑖000100𝑖 (50)

Algorithm 15. (1) Input 𝐴₁,𝐴₂,…,𝐴_𝑝, 𝐶₁,𝐶₂,…,𝐶_𝑝, 𝐵₁,…,𝐵_𝑞, 𝐷₁,…,𝐷_𝑞;
(2) compute 𝐴_𝑘1, 𝐴_𝑘2, 𝐵_𝑙1, 𝐵_𝑙2, 𝐶_𝑘1, 𝐶_𝑘2, 𝐷_𝑙1, 𝐷_𝑙2, 𝑆, 𝐺 by (18), (19), (20), (24), and (25);
(3) make the SVD of 𝑆 with the form of (27) and compute 𝑊₁, 𝑊₂, 𝑉₁, 𝑉₂ by (28);
(4) if (37) holds, continue, or go to step 6;
(5) input 𝐸∈𝒞^𝑛×𝑛, compute the solution to Problem 2 with 𝒮=𝐻𝑅_𝑐𝒞^𝑛×𝑛 by (40);
(6) input 𝐹∈𝒞^𝑛×𝑛, compute the solution of Problem 1 with 𝒮=𝐻𝑅_𝑐𝒞^𝑛×𝑛 by (32).

Example 1. We consider the case of 𝑝=1 and 𝑞=1. Suppose 𝐴,𝐶∈𝒞^5×6, 𝐵,𝐷∈𝒞^6×4, and𝐴=^{−2.2163+1.2010𝑖2.2524−1.1328𝑖0−0.0955+1.3420𝑖4.3215−2.3145𝑖1.2120+1.9804𝑖−1.16253.2510𝑖02.1800−0.3125𝑖−0.2235+1.4140𝑖5.7071}_{1.0500−2.0120𝑖2.4020+1.2500𝑖03.2106𝑖−1.60802.25600000003.2000−4.1955+1.2000𝑖01.1025−0.2400𝑖−0.8902−0.2408𝑖1.2010+0.7071𝑖},𝐶=^{4.1365+2.6614𝑖1.5828−0.2259𝑖2.2091+1.5678𝑖2.4154𝑖−1.8939+0.4692𝑖2.5155−0.7604𝑖1.0118−0.2844𝑖−3.4235+1.5798𝑖1.3798−0.7942𝑖1.3050+2.9900𝑖−2.7833+0.0592𝑖1.8744−0.7889𝑖}_{0000002.1991−1.0648𝑖−1.5298𝑖5.6808+1.6029𝑖−4.0150+2.1200𝑖2.4387−1.9600𝑖3.4200−2.6288𝑖2.98832.4329−0.6405𝑖1.4611−1.0503𝑖3.7680−1.4980𝑖−2.0266−0.9334𝑖1.5756𝑖},𝐵=,0.5200+0.7200𝑖−0.20000.4500+0.3600𝑖0−0.3394−0.5091𝑖0.1414−0.3536𝑖−0.3200−0.6500𝑖00.5940+0.2970𝑖0.0707+0.0354𝑖0.070000.0200𝑖0.25000.3500+0.8400𝑖0−0.3960−0.5091𝑖0.1414+0.3536𝑖−0.3200+0.2400𝑖00.5940+0.2970𝑖0.2121+0.0354𝑖0.07070𝐷=^{−0.3140+0.1069𝑖0.4954−0.0615𝑖0.7469+0.4876𝑖−0.0662+0.0817𝑖0.4255−0.5754𝑖0.0348+0.5182𝑖−0.2522+0.5751𝑖−0.0105−0.0928𝑖0.0883+0.0649𝑖0.2201+0.0182𝑖0.3133+0.3271𝑖0.0076−0.0138𝑖}_{0.8675+0.7972𝑖−0.3105−0.0837𝑖−0.1461+0.2541𝑖0.2620+0.0983𝑖−0.4591+0.6268𝑖−0.0405−0.6447𝑖0.1517−0.6198𝑖0.0256+0.2057𝑖0.1439−0.2759𝑖0.4722+0.1091𝑖0.4952+0.3680𝑖−0.0414−0.1178𝑖}. (51)We can easily verify the solvability condition (37) does not holds. For given 𝐸∈𝒞^6×6,𝐸=^{0.4424+0.6274𝑖−0.1765−0.0108𝑖0.4006+0.3308𝑖0.3069−0.1505+0.1262𝑖1.200+0.0045𝑖0.6374−0.0108𝑖0.2246−0.2738𝑖0.1509−0.4327𝑖−0.0434−0.0915𝑖0.1385−0.1800𝑖0−0.4950−0.4172𝑖0.1414−0.0354𝑖−0.3818−0.1750𝑖−0.2475+0.0619𝑖−0.1630−0.0540𝑖0.2032𝑖}_{−0.5129−0.4025𝑖0.1956−0.0157𝑖0.0868+0.5565𝑖−0.1227−0.2382−0.0911𝑖00.1878+0.3187𝑖0.1789+0.3334𝑖0.0271+0.7668𝑖−0.0434+0.0915𝑖−0.1024−0.0002+0.0150𝑖−0.2404−0.6010𝑖0.1414+0.0354𝑖−0.2546−0.3341𝑖−0.2475−0.0619𝑖−0.0002−0.0150𝑖0.0256}, (52)applying Algorithm 15, we get the following results:_𝑋=^{−0.1949+0.0000𝑖−0.0395+0.3097𝑖0.1067+0.0353𝑖0.6067−0.2888𝑖0.0380−0.6272𝑖−0.1218+0.1242𝑖−0.0395−0.3097𝑖−0.7354+0.0000𝑖0.0606−0.1460𝑖−0.1095+0.4086𝑖0.6954−0.0593𝑖0.0380−0.6272𝑖0.1067−0.0353𝑖0.0606+0.1460𝑖0.00090.39570.0606−0.1460𝑖0.1067+0.0353𝑖}_{0.6067+0.2888𝑖−0.1095−0.4086𝑖0.3957−0.1562−0.1095+0.4086𝑖0.6067−0.2888𝑖0.0380+0.6272𝑖0.6954+0.0593𝑖0.0606+0.1460𝑖−0.1095−0.4086𝑖−0.7354−0.0000𝑖−0.0395+0.3097𝑖−0.1218−0.1242𝑖0.0380+0.6272𝑖0.1067−0.0353𝑖0.6067+0.2888𝑖−0.0395−0.3097𝑖−0.1949−0.0000𝑖},𝐴𝑋−𝐶²+𝑋𝐵−𝐷²=min_{𝑋∈𝐻𝑅𝑐ℂ6×6}‖𝐴𝑋−𝐶‖²+‖𝑋𝐵−𝐷‖²𝑋=160.0357,𝑋−^∗=4.1192×10⁻¹⁶,𝑋−𝑅𝑋𝑅=0,𝑋−𝐸=3.7397. (53)(54)

The following example is about Problem 2 with 𝒮=𝐻𝑅_𝑐𝒞^6×6, 𝑝=1 and 𝑞=0. We list results of comparison of the solutions computed by Algorithm 15 and MATLAB procedure 𝑋=𝐴⧵𝐶.

Example 2. Let 𝑊_𝑎∈𝒞^8×8, 𝑉_𝑎∈𝒞^6×6 be unitary matrices,𝑊_𝑎=,𝑉−0.1590+0.2812𝑖−0.1803−0.1822𝑖0.4873+0.0557𝑖0.1272+0.2856𝑖∶0.1159−0.1077𝑖0.4983−0.2783𝑖0.4127−0.2359𝑖0.3195+0.2342𝑖∶0.2197−0.1364𝑖0.1146−0.1466𝑖0.2900+0.0440𝑖−0.5309−0.0423𝑖∶0.0016−0.0047𝑖0.0108−0.0047𝑖−0.0257+0.0018𝑖−0.0181+0.0126𝑖∶0.2576−0.0271𝑖0.2935+0.2509𝑖−0.3014+0.0304𝑖0.3562+0.0778𝑖∶0.4283+0.1012𝑖0.0858−0.1579𝑖−0.1758−0.3913𝑖−0.5198+0.0316𝑖∶0.3884−0.2103𝑖0.2167+0.2285𝑖0.1236−0.0447𝑖0.1348−0.0089𝑖∶0.5625+0.1774𝑖−0.2720−0.4682𝑖−0.0047+0.3918𝑖0.1951+0.0273𝑖∶−0.5270−0.1794𝑖0.1734−0.2471𝑖0.2466+0.1275𝑖0.0437+0.1077𝑖0.0562−0.0163𝑖−0.2264+0.4255𝑖−0.1461+0.0547𝑖−0.0662−0.0426𝑖−0.1748+0.4297𝑖0.4788+0.1119𝑖−0.1543−0.2114𝑖0.0422−0.0388𝑖−0.0011−0.0201𝑖0.0249+0.0695𝑖0.0506+0.2945𝑖0.7638−0.5658𝑖−0.3766−0.0582𝑖0.3605−0.2131𝑖−0.4447+0.1694𝑖−0.0868−0.0677𝑖−0.1860−0.3130𝑖−0.2654−0.1257𝑖0.0594+0.2924𝑖−0.0976+0.0376𝑖0.3112−0.2036𝑖0.3628−0.1215𝑖0.5860−0.0319𝑖0.1016+0.1755𝑖0.2087−0.1071𝑖−0.0638−0.1313𝑖−0.1736−0.2252𝑖0.0555−0.0788𝑖_𝑎=^{0.46210.1052−0.6218−0.1836−0.58970.0852−0.2474−0.0088𝑖−0.1896−0.5726𝑖0.1951−0.1974𝑖−0.5870−0.1096𝑖−0.2811+0.1327𝑖−0.2113−0.0036𝑖0.4418−0.0309𝑖−0.2001−0.2357𝑖0.0004+0.2178𝑖0.0696+0.1631𝑖0.1926−0.3664𝑖−0.6629−0.1362𝑖}_{−0.1299−0.4383𝑖0.1856+0.1319𝑖0.1214−0.4193𝑖0.0858+0.5057𝑖−0.2366−0.1016𝑖−0.0917−0.4592𝑖−0.2390+0.2258𝑖−0.1938−0.3803𝑖0.0533+0.3571𝑖0.3007+0.3102𝑖−0.3167−0.3755𝑖0.3810−0.0794𝑖0.4263+0.1739𝑖0.3926−0.3760𝑖0.3391−0.2301𝑖0.2360+0.2699𝑖−0.0153+0.2763𝑖0.0807+0.3359𝑖}, (55)and 𝐷(𝑡)=^𝐷_zeros(2,6), where 𝐷=diag(3,2,1,𝑡/1,𝑡/2,𝑡/3) and zeros(2,6) is a 2×6 zeros matrix. Let 𝐴(𝑡)=𝑊_𝑎𝐷(𝑡)𝑉^∗_𝑎 and 𝑋∈𝐻𝑅_𝑐𝒞^6×6,𝑋=^{0.2543−0.3625−1.1647𝑖−0.1609−0.2793𝑖0.8839−0.4914𝑖0.5675−0.0043𝑖−0.4143−0.4378𝑖−0.3625+1.1647𝑖−1.38860.0672−0.6470𝑖−0.2316+0.2397𝑖0.0524−0.4418𝑖0.5675−0.0043𝑖−0.1609+0.2793𝑖0.0672+0.6470𝑖−0.4312−0.25000.0672−0.6470𝑖−0.1609−0.2793𝑖}_{0.8839+0.4914𝑖−0.2316−0.2397𝑖−0.2500−0.0380−0.2316+0.2397𝑖0.8839−0.4914𝑖0.5675+0.0043𝑖0.0524+0.4418𝑖0.0672+0.6470𝑖−0.2316−0.2397𝑖−1.3886−0.3625−1.1647𝑖−0.4143+0.4378𝑖0.5675+0.0043𝑖−0.1609+0.2793𝑖0.8839+0.4914𝑖−0.3625+1.1647𝑖0.2543}. (56)Then compute 𝐶(𝑡)=𝐴(𝑡)𝑋 for 𝑡=10,1,10⁻¹,10⁻²,…,10⁻¹³. Obviously, Problem 2 with 𝒮=𝐻𝑅_𝑐𝒞^𝑛×𝑛 is consistent for each value of 𝑡. For matrices 𝐴, 𝐶 obtained above, we first use Algorithm 15 to obtain the Hermitian 𝑅-conjugate solutions approximate to 𝑋, then compute the solutions of 𝐴𝑋=𝐶 by MATLAB procedure 𝑋=𝐴⧵𝐶. Let 𝑋_𝑖𝑠 denote the solutions computed by Algorithm 15 and 𝑋_𝑖𝑠 the solutions by MATLAB procedure 𝑋=𝐴⧵𝐶. Let 𝑒1=𝐴𝑋_𝑖−𝐶,𝑒2=𝑋_𝑖−𝑋,𝑒3=𝑋_𝑖−𝑅𝑋_𝑖𝑅,𝑒4=𝑋^𝑇_𝑖−𝑋_𝑖,𝑡𝑒1=𝐴𝑋_𝑖−𝐶,𝑡𝑒2=𝑋_𝑖−𝑋,𝑡𝑒3=𝑋_𝑖−𝑅𝑋_𝑖𝑅,𝑡𝑒4=𝑋^𝑇_𝑖−𝑋_𝑖. (57)

Analysis of Results. As a general observation from Table 1, we find that the performance of Algorithm 15 to solve Problem 2 is very good and that of the MATLAB procedure 𝑋=𝐴⧵𝐶 is quite sensitive to the conditioning of matrix 𝐴. For 𝑡=10,1,…,10⁻⁴, both methods behave well. In this case we should choose MATLAB procedure 𝑋=𝐴⧵𝐶 to solve Problem 2 with 𝒮=𝐻𝑅_𝑐𝒞^𝑛×𝑛 for it is simple. However, as some singular values of 𝐴 are close to zero, the solutions 𝑋_𝑖 computed by MATLAB procedure 𝑋=𝐴⧵𝐶 do not satisfy 𝐴𝑋=𝐶 and gradually lose the property of Hermitian 𝑅-conjugate matrix, while Algorithm 15 does it well. Hence, when 𝐴 has small singular values close to zero, the Algorithm 15 predominates over MATLAB procedure 𝑋=𝐴⧵𝐶.

Table 1 

Comparison of results by Algorithm 15 and the MATLAB procedure 𝐴⧵𝐶.

6. Conclusion

In this paper, we converted the Hermitian 𝑅-conjugate generalized Procrustes problem to real symmetric Procrustes problem trickly and obtained its solution set. We also investigated the Hermitian 𝑅-conjugate solution to the linear system of complex matrix equations 𝐴₁𝑋=𝐶₁, 𝐴₂𝑋=𝐶₂,…,𝐴_𝑝𝑋=𝐶_𝑝, 𝑋𝐵₁=𝐷₁,…,𝑋𝐵_𝑞=𝐷_𝑞 and established solvable conditions and the formula for the Hermitian 𝑅-conjugate solution. Moreover, we showed the optimal approximation solution to a given matrix in the above two corresponding solution set is unique, respectively. As applications, a numerical algorithm has been given and the examples have illustrated the feasibility of the algorithm. Additionally, we can further consider the least square Hermitian 𝑅-conjugate solution of the system 𝐴_𝑘𝑋𝐴^∗_𝑘=𝐶_𝑘, 𝑘=1,2,…,𝑛 (𝑛 is positive integer) and the corresponding optimal approximation problem.

Acknowledgments

The authors are very much indebted to the anonymous referees for their constructive and valuable comments and suggestions which greatly improved the original paper. This research was supported by the Innovation Foundation of Shanghai Municipal Education Commission (13ZZ080), the National Natural Science Foundation of China (11391240185), the Shanghai University Scientific Selection and Cultivation for Outstanding Young Teachers in special fund (sjr10009), and the Natural Science Foundation of Guangxi Province (no. 2012GXNSFBA053006).