Eigenvector-Free Solutions to the Matrix Equation 𝐴𝑋𝐵^𝐻=𝐸 with Two Special Constraints

Abstract

The matrix equation 𝐴𝑋𝐵^𝐻=𝐸 with 𝑆𝑋=𝑋𝑅 or 𝑃𝑋=𝑠𝑋𝑄 constraint is considered, where S, R are Hermitian idempotent, P, Q are Hermitian involutory, and 𝑠=±1. By the eigenvalue decompositions of S, R, the equation 𝐴𝑋𝐵^𝐻=𝐸 with 𝑆𝑋=𝑋𝑅 constraint is equivalently transformed to an unconstrained problem whose coefficient matrices contain the corresponding eigenvectors, with which the constrained solutions are constructed. The involved eigenvectors are released by Moore-Penrose generalized inverses, and the eigenvector-free formulas of the general solutions are presented. By choosing suitable matrices S, R, we also present the eigenvector-free formulas of the general solutions to the matrix equation 𝐴𝑋𝐵^𝐻=𝐸 with 𝑃𝑋=𝑠𝑋𝑄 constraint.

1. Introduction

In [1], Chen has denoted a square matrix 𝑋, the reflexive or antireflexive matrix with respect to 𝑃 by 𝑃𝑋=𝑋𝑃or𝑃𝑋=−𝑋𝑃, (1) where the matrix 𝑃∈𝒞^𝑛×𝑛 is Hermitian involutory. He also pointed out that these matrices possessed special properties and had wide applications in engineering and scientific computations [1, 2]. So, solving the matrix equation or matrix equations with these constraints is maybe interesting [3–14]. In this paper, we consider the matrix equation 𝐴𝑋𝐵^𝐻=𝐸 (2) with constraint 𝑃𝑋=𝑠𝑋𝑄or𝑆𝑋=𝑋𝑅, (3) where the matrices 𝐴∈𝒞^𝑚×𝑛, 𝐵∈𝒞^𝑝×𝑛, 𝐸∈𝒞^𝑚×𝑝, the Hermitian involutory matrices 𝑃,𝑄∈𝒞^𝑛×𝑛, the Hermitian idempotent matrices 𝑆,𝑅∈𝒞^𝑛×𝑛, and the scalars 𝑠=±1.

Equation (2) with different constraints such as symmetry, skew-symmetry, and 𝑃𝑋=±𝑋𝑃, was discussed in [9–11, 15–21], where existence conditions and the general solutions to the constrained equation were presented. By generalized singular value decomposition (GSVD) [22, 23], the authors of [15–17] simplified the matrix equation by diagonalizing the coefficient matrices and block-partitioned the new variable matrices into several block matrices, then imposed the constrained condition on subblocks, and determined the unknown subblocks separately for (2) with symmetric constraint. A similar strategy was also used in [18]; the authors achieved symmetric, skew-symmetric, and positive semidefinite solutions to (2) by quotient singular value decomposition (QSVD) [24, 25]. Moreover, in [20], CCD [26] was used for establishing a formula of the general solutions to (2) with diagonal constraint.

In [19], we have presented an eigenvector-free solution to the matrix equation (2) with constraint 𝑃𝑋=±𝑋𝑃, where we represented its general solution and existence condition by 𝑔-inverses of the matrices 𝐴, 𝐵, and 𝑃. Note that the 𝑔-inverses are always not unique, and they can be generalized to the Moore-Penrose generalized inverses. Moreover, the constraint which guarantees the eigenvector-free expressions can be maybe improved further. So, in this paper, we focus on (2) with generalized constraint 𝑃𝑋=𝑠𝑋𝑄 or another constraint 𝑆𝑋=𝑋𝑅; our ideas are based on the following observations. (1)If we set 1𝑆=2()1𝐼+𝑃,𝑅=2(𝐼+𝑠𝑄), (4) then 𝑆 and 𝑅 are both Hermitian idempotent. The above fact implies 𝑃𝑋=𝑠𝑋𝑄 is the special case of 𝑆𝑋=𝑋𝑅. So, we only discuss (2) with 𝑆𝑋=𝑋𝑅 constraint and construct the 𝑃𝑋=𝑠𝑋𝑄 constrained solution by selecting suitable matrices 𝑅, 𝑄 as (4). (2)With the eigenvalue decompositions (EVDs) of the Hermitian matrices 𝑅, 𝑆, matrix 𝑋 with 𝑆𝑋=𝑋𝑅 constraint can be rewritten in (lower dimensional) two free variables 𝑋 and 𝑌. And the corresponding constrained problem can be equivalently transformed to an unconstrained equation 𝐴₁𝑋𝐵^𝐻₁+𝐴₂𝑌𝐵^𝐻₂=𝐸, (5) with given coefficient matrices 𝐴_𝑖, 𝐵_𝑖, 𝑖=1,2 (one can see the details of this discussion in Section 2). (3)The general solutions and existence conditions of (5) can be represented by the Moore-Penrose generalized inverses of 𝐴_𝑖, 𝐵_𝑖, 𝑖=1,2 [15, 20, 27–29]. However, the formulas above are maybe not simpler because the coefficient matrices contain the eigenvectors of 𝑆, 𝑅. In fact, the Hermitian idempotence of the matrices 𝑆, 𝑅 implies they only have two clusters different eigenvalues, and their corresponding eigenvectors appear in the expression of general solutions, and existence conditions can be easily represented by 𝑆, 𝑅 themselves. So we present a simple and eigenvector-free formulation for the constrained general solution.

The rest of this paper is organized as follows. In Section 2, we give the general solutions and the existence condition to (2) with 𝑆𝑋=𝑋𝑅 constraint by the EVDs of 𝑆, 𝑅. In Section 3, we present the corresponding eigenvector-free representations. Equation (2) with 𝑃𝑋=𝑠𝑋𝑄 constraint is regarded as the special case of (2) with 𝑆𝑋=𝑋𝑅 constraint, and its eigenvector-free representation is given in Section 4. Numerical examples are given in Section 5 to display the effectiveness of our theorems.

We will use the following notations in the rest of this paper. Let 𝒞^𝑚×𝑛 denote the space of complex 𝑚×𝑛 matrix. For a matrix 𝐴, 𝐴^𝐻 and 𝐴^† denote its transpose and Moore-Penrose generalized inverse, respectively. Matrix 𝐼_𝑛 is identity matrix with order 𝑛; 𝑂_𝑚×𝑛 refers to 𝑚×𝑛 zero matrix, and 𝑂_𝑛 is the zero matrix with order 𝑛. For any matrix 𝐴∈𝒞^𝑚×𝑛, we also denote 𝒫_𝐴=𝐴𝐴^†,𝐾_𝐴=𝐼_𝑚−𝒫_𝐴. (6) So, 𝒫_𝐴^𝐻=𝐴^†𝐴,𝐾_𝐴^𝐻=𝐼_𝑛−𝒫_𝐴^𝐻. (7)

2. Solution to (2) with 𝑆𝑋=𝑅𝑋 Constraint by the EVDs

For the Hermitian idempotent matrices 𝑆, 𝑅, let 𝑆=𝑈diag𝐼_𝑘,𝑂_𝑛−𝑘𝑈^𝐻,𝑅=𝑉diag𝐼_𝑙,𝑂_𝑛−𝑙𝑉^𝐻 (8) be their two eigenvalue decompositions with unitary matrices 𝑈, 𝑉, respectively. Then 𝑆𝑋=𝑋𝑅 holds if and only if diag𝐼_𝑘,𝑂_𝑛−𝑘𝑋=𝑋diag𝐼_𝑙,𝑂_𝑛−𝑙, (9) where 𝑋=𝑈^𝐻𝑋𝑉. And the constrained solution 𝑋 can be expressed in 𝑋=𝑈diag𝑋,𝑌𝑉^𝐻,𝑋∈𝒞^𝑘×𝑙,𝑌∈𝒞^{(𝑛−𝑘)×(𝑛−𝑙)}. (10) Partitioning 𝑈=[𝑈₁,𝑈₂], 𝑉=[𝑉₁,𝑉₂] and using the transformations (10), (2) with 𝑆𝑋=𝑋𝑅 constraint is equivalent to the following unconstrained problem: 𝐴₁𝑋𝐵^𝐻₁+𝐴₂𝑌𝐵^𝐻₂=𝐸, (11) where 𝐴₁=𝐴𝑈₁,𝐵₁=𝐵𝑉₁,𝐴₂=𝐴𝑈₂,𝐵₂=𝐵𝑉₂. (12)

For the unconstrained problem (11), we introduce the results about its existence conditions and expression of solutions.

Lemma 1. Given 𝐴∈𝒞^𝑚×𝑛, 𝐵∈𝒞^𝑝×𝑞, 𝐶∈𝒞^𝑚×𝑟, 𝐷∈𝒞^𝑠×𝑞, and 𝐸∈𝒞^𝑚×𝑞, the linear matrix equation 𝐴𝑋𝐵+𝐶𝑌𝐷=𝐸 is consistent if and only if 𝒫_𝐺𝐾_𝐴𝐸𝒫_𝐷^𝐻=𝐾_𝐴𝐸,𝒫_𝐶𝐸𝐾_𝐵^𝐻𝒫_𝐽^𝐻=𝐸𝐾_𝐵^𝐻, (13) or, equivalently, if and only if 𝐾_𝐺𝐾_𝐴𝐸=0,𝐾_𝐴𝐸𝐾_𝐷^𝐻𝐾=0,_𝐶𝐸𝐾_𝐵^𝐻=0,𝐸𝐾_𝐵^𝐻𝐾_𝐽^𝐻=0, (14) where 𝐺=𝐾_𝐴𝐶 and 𝐽=𝐷𝐾_𝐵^𝐻. And a representation of the general solution is 𝑌=𝐺^†𝐾_𝐴𝐸𝐷^†+𝑇−𝒫_𝐺^𝐻𝑇𝒫_𝐷,𝑋=𝐴^†(𝐸−𝐶𝑌𝐷)𝐵^†+𝑍−𝒫_𝐴^𝐻𝑍𝒫_𝐵, (15) with 𝑇=𝐶𝐾_𝐺^𝐻†𝐼_𝑚−𝐶𝐺^†𝐾_𝐴𝐸𝐾_𝐵^𝐻𝐽^†+𝑊−𝒫_{(𝐶𝐾𝐺𝐻)}^𝐻𝑊𝒫_𝐽, (16) where the matrices 𝑊∈𝒞^𝑟×𝑠 and 𝑍∈𝒞^𝑛×𝑝 are arbitrary.

The lemma is easy to verify; we can turn to [27] for details. The difference between them is that we replace the 𝑔-inverse in the theorem of [27] by the corresponding Moore-Penrose generalized inverse, and the expression of solutions is complicated relatively. However, compared with the multiformity of the 𝑔-inverses, the Moore-Penrose generalized inverse involved representation is unique and fixed.

Apply Lemma 1 on the unconstrained problem (11), we have the following theorem.

Theorem 2. The matrix equation 𝐴𝑋𝐵^𝐻=𝐸 with constraint 𝑆𝑋=𝑋𝑅 is consistent if and only if 𝒫𝐺𝐾𝐴₁𝐵𝐸𝒫₂𝐴=𝐾₁𝐴𝐸,𝒫₂𝐵𝐸𝐾₁𝒫̂𝐽^𝐻𝐵=𝐸𝐾₁, (17) where 𝐴𝐺=𝐾₁𝐴₂,𝐽̂=𝐵^𝐻₂𝐾𝐵₁. (18) In the meantime, a general solution is given by 𝑌=𝐺^†𝐾𝐴₁𝐸𝐵^𝐻₂^†+𝐴₂𝐾𝐺^𝐻†𝐼_𝑚−𝐴₂𝐺^†𝐾𝐴₁𝐵𝐸𝐾₁𝐽^†̂𝐺−𝒫^𝐻𝐴₂𝐾𝐺^𝐻†𝐼_𝑚−𝐴₂𝐺^†𝐾𝐴₁𝐵𝐸𝐾₁𝐽^†̂𝒫𝐵^𝐻₂𝐺+𝑊−𝒫^𝐻𝐵𝑊𝒫^𝐻₂−𝒫₍𝐴_2𝐾𝐺^𝐻)𝐻𝐽̂𝐺𝑊𝒫+𝒫^𝐻𝒫₍𝐴_2𝐾𝐺^𝐻)𝐻𝑊𝒫𝐽𝒫̂𝐵^𝐻₂,𝑋=𝐴^†₁𝐸−𝐴₂𝑌𝐵^𝐻₂𝐵^𝐻†₁𝐴+𝑍−𝒫^𝐻₁𝐵𝑍𝒫^𝐻₁, (19) where the matrices 𝑊 and 𝑍 are arbitrary.

In order to separate 𝑌 from 𝑋 of the second equality in (19), we substitute 𝑌 into 𝑋. Let 𝑌_∗=𝐺^†𝐾𝐴₁𝐸𝐵^𝐻₂^†+𝐴₂𝐾𝐺^𝐻†𝐼_𝑚−𝐴₂𝐺^†𝐾𝐴₁𝐵𝐸𝐾₁𝐽^†̂𝐺−𝒫^𝐻𝐴₂𝐾𝐺^𝐻†𝐼_𝑚−𝐴₂𝐺^†𝐾𝐴₁𝐵𝐸𝐾₁𝐽^†̂𝒫𝐵^𝐻₂,𝑋_∗=𝐴^†₁𝐸𝐵^𝐻₁^†−𝐴^†₁𝐴₂𝑌_∗𝐵^𝐻₂𝐵^𝐻†₁, (20) together with 𝐵^†₂𝐵₂𝐵^𝐻₂=𝐵^†₂𝐵₂^𝐻𝐵^𝐻₂=𝐵^𝐻₂,𝐴₂𝐾𝐺^𝐻𝐴₂𝐾𝐺^𝐻†𝐴₂𝐾𝐺^𝐻=𝐴₂𝐾𝐺^𝐻. (21) Then (19) can be rewritten as 𝑌=𝑌_∗𝐺+𝑊−𝒫^𝐻𝐵𝑊𝒫^𝐻₂−𝒫₍𝐴_2𝐾𝐺^𝐻)𝐻𝐽̂𝐺𝑊𝒫+𝒫^𝐻𝒫₍𝐴_2𝐾𝐺^𝐻)𝐻𝑊𝒫𝐽𝒫̂𝐵^𝐻₂,𝑋=𝑋_∗𝐴+𝑍−𝒫^𝐻₁𝐵𝑍𝒫^𝐻₁−𝐴^†₁𝐴₂𝐾𝐺^𝐻𝐽̂𝑊𝐾𝐵^𝐻₂𝐵^𝐻†₁. (22)

3. Eigenvector-Free Formulas of the General Solutions to (2) with 𝑆𝑋=𝑋𝑅 Constraint

The existence conditions and the expression of the general solution given in Theorem 2 contain the eigenvector matrices of 𝑆, 𝑅, respectively. This implies that the eigenvalue decompositions will be included. In this section, we intend to release the involved eigenvectors in detailed expressions. With the first equality in (8), we have 𝑈₁𝑈^𝐻₁=𝑆,𝑈₂𝑈^𝐻₂=𝐼_𝑛𝑉−𝑆,₁𝑉^𝐻₁=𝑅,𝑉₂𝑉^𝐻₂=𝐼_𝑛−𝑅. (23) Note that 𝑈_𝑖(𝐴𝑈_𝑖)^† is the Moore-Penrose generalized inverse of 𝐴𝑈_𝑖𝑈^𝐻_𝑖, which gives 𝒫𝐴_𝑖=𝐴_𝑖𝐴^†_𝑖=𝐴𝑈_𝑖𝑈^𝐻_𝑖𝐴𝑈_𝑖𝑈^𝐻_𝑖^†=𝐴_𝑖𝐴^†_𝑖=𝒫_𝐴𝑖, (24) where 𝐴₁=𝐴𝑈₁𝑈^𝐻₁=𝐴𝑆,𝐴₂=𝐴𝑈₂𝑈^𝐻₂=𝐴𝐼_𝑛−𝑆. (25) Then 𝐾𝐴_𝑖=𝐼_𝑚𝐴−𝒫_𝑖=𝐼_𝑚−𝒫_𝐴𝑖=𝐾_𝐴𝑖,𝐺𝑈^𝐻₂=𝐾_𝐴1𝐴₂. (26) Set 𝐵₁=𝐵𝑉₁𝑉^𝐻₁=𝐵𝑅,𝐵₂=𝐵𝑉₂𝑉^𝐻₂=𝐵𝐼_𝑛−𝑅, (27) and denote 𝐺=𝐾_𝐴1𝐴₂,𝐽=𝐵^𝐻₂𝐾_𝐵1. (28) It is not difficult to verify that 𝑉₂𝐽̂=𝐽,𝐺𝑈^𝐻₂=𝐺, (29) together with 𝒫𝐺=𝐺𝑈^𝐻₂𝐺𝑈^𝐻₂^†=𝒫_𝐺,𝒫̂𝐽^𝐻=𝑉₂𝐽̂^†𝑉₂𝐽̂=𝒫_𝐽^𝐻. (30) Then the first equality of (17) can be rewritten as 𝒫_𝐺𝐾_𝐴1𝐸𝒫_𝐵2=𝐾_𝐴1𝐸, (31) and the other can be rewritten as 𝒫_𝐴2𝐸𝐾_𝐵1𝒫_𝐽^𝐻=𝐸𝐾_𝐵1. (32) Now, we consider the simplification of the general solution 𝑋 given by (10), which can be rewritten as 𝑋=𝑈₁𝑋𝑉^𝐻₁+𝑈₂𝑌𝑉^𝐻₂. (33) Note that 𝑈₂𝐺^†=𝐺𝑈^𝐻₂^†=𝐺^†𝐺,𝐾^𝐻𝑈^𝐻₂=𝑈^𝐻₂𝐾_𝐺^𝐻,𝑈₂𝐴^†₂=𝐴^†₂. (34) Together with (26), 𝑈₂𝑌_∗𝑉^𝐻₂=𝑈₂𝐺^†𝐾𝐴₁𝐸𝐵^𝐻₂^†+𝐴₂𝐾𝐺^𝐻†×𝐼_𝑚−𝐴₂𝐺^†𝐾𝐴₁𝐵𝐸𝐾₁𝐽^†̂𝐺−𝒫^𝐻𝐴₂𝐾𝐺^𝐻†×𝐼_𝑚−𝐴₂𝐺^†𝐾𝐴₁𝐵𝐸𝐾₁𝐽^†̂𝒫𝐵^𝐻₂𝑉^𝐻₂=𝐺^†𝐾_𝐴1𝐸𝐵^𝐻₂^†+𝐴₂𝐾_𝐺^𝐻†×𝐼_𝑚−𝐴₂𝐺^†𝐾_𝐴1𝐸𝐾_𝐵1𝐽^†−𝒫_𝐺^𝐻𝐴₂𝐾_𝐺^𝐻†×𝐼_𝑚−𝐴₂𝐺^†𝐾_𝐴1𝐸𝐾_𝐵1𝐽^†𝒫_𝐵^𝐻₂, (35) so we can represent 𝑈_𝑗2𝑌_∗𝑉^𝐻₂ by a given expression of 𝐴_𝑖, 𝐵_𝑖, 𝐸. Let 𝑓𝐴₁,𝐴₂,𝐵₁,𝐵₂,𝐸=𝐺^†𝐾_𝐴1𝐸𝐵^𝐻₂^†+𝐴₂𝐾_𝐺^𝐻†×𝐼_𝑚−𝐴₂𝐺^†𝐾_𝐴1𝐸𝐾_𝐵1𝐽^†−𝒫_𝐺^𝐻𝐴₂𝐾_𝐺^𝐻†×𝐼_𝑚−𝐴₂𝐺^†𝐾_𝐴1𝐸𝐾_𝐵1𝐽^†𝒫_𝐵^𝐻₂. (36) Hence, we have 𝑈_𝑗2𝑌_∗𝑉^𝐻₂=𝑓𝐴₁,𝐴₂,𝐵₁,𝐵₂,𝑈,𝐸₁𝑋_∗𝑉^𝐻₁=𝐴^†₁𝐸𝐵^𝐻₁^†−𝐴^†₁𝐴₂𝑈₂𝑌_∗𝑉^𝐻₂𝐵^𝐻₂𝐵^𝐻₁^†=𝐴^†₁𝐸𝐵^𝐻₁^†−𝐴^†₁𝐴₂𝑓𝐴₁,𝐴₂,𝐵₁,𝐵₂,𝐸𝐵^𝐻₂𝐵^𝐻₁^†. (37) Since 𝑉₂𝐾̂𝐽=𝑉₂𝐼_𝑛−𝑙̂𝐽−𝒫=𝐼_𝑝−𝑉₂𝐽̂𝑉₂𝐽̂^†𝑉₂=𝐾_𝐽𝑉₂, (38) then 𝑈₁𝐴𝑍−𝒫^𝐻₁𝐵𝑍𝒫^𝐻₁−𝐴^†₁𝐴₂𝐾𝐺^𝐻𝐽̂𝑊𝐾𝐵^𝐻₂𝐵^𝐻†₁𝑉^𝐻₁=𝑈₁𝑍𝑉^𝐻₁−𝒫_𝐴^𝐻₁𝑈₁𝑍𝑉^𝐻₁𝒫_𝐵^𝐻₁−𝐴^†₁𝐴₂𝐾_𝐺^𝐻𝑈₂𝑊𝑉^𝐻₂𝐾_𝐽𝐵^𝐻₂𝐵^𝐻₁^†,𝑈₂𝐺𝑊−𝒫^𝐻𝐵𝑊𝒫^𝐻₂−𝒫₍𝐴₂^𝐾𝐺^𝐻)𝐻𝐽̂𝐺𝑊𝒫+𝒫^𝐻𝒫₍𝐴₂^𝐾𝐺^𝐻)𝐻𝑊𝒫𝐽𝒫̂𝐵^𝐻₂𝑉^𝐻₂=𝑈₂𝑊𝑉^𝐻₂−𝒫_𝐺^𝐻𝑈₂𝑊𝑉^𝐻₂𝒫_𝐵^𝐻₂−𝒫_{(𝐴2𝐾𝐺𝐻)}^𝐻𝑈₂𝑊𝑉^𝐻₂𝒫_𝐽+𝒫_𝐺^𝐻𝒫_{(𝐴2𝐾𝐺𝐻)}^𝐻𝑈₂𝑊𝑉^𝐻₂𝑃_𝐽𝒫_𝐵^𝐻₂. (39) Letting 𝑈₁𝑍𝑉^𝐻₁+𝑈₂𝑊𝑉^𝐻₂=𝐹, (40) it is not difficult for us to verify 𝑆𝐹=𝐹𝑅. Together with 𝐴₂𝑈₁=0,𝐴₁𝑈₂=0,𝑉^𝐻₂𝐵^†₁=0,𝑉^𝐻₁𝐵^†₂=0, (41) the following equality holds: 𝑃_𝐴^𝐻₁𝑈₁𝑍𝑉^𝐻₁𝒫_𝐵^𝐻₁+𝒫_𝐺^𝐻𝑈₂𝑊𝑉^𝐻₂𝒫_𝐵^𝐻₂=𝑃_𝐴^𝐻₁+𝒫_𝐺^𝐻𝑈₂𝑊𝑉^𝐻₂+𝑈₁𝑍𝑉^𝐻₁×𝒫_𝐵^𝐻₁+𝒫_𝐵^𝐻₂=𝑃_𝐴^𝐻₁+𝒫_𝐺^𝐻𝐹𝒫_𝐵^𝐻₁+𝒫_𝐵^𝐻₂. (42) Note that 𝐺𝑈₁=0,𝐴₂𝐾_𝐺^𝐻𝑈₁=0. (43) Then 𝐴₂𝐾_𝐺^𝐻𝑈₂𝑊𝑉^𝐻₂=𝐴₂𝐾_𝐺^𝐻𝑈₂𝑊𝑉^𝐻₂+𝑈₁𝑍𝑉^𝐻₁=𝐴₂𝐾_𝐺^𝐻𝐹. (44) Hence, 𝐴^†₁𝐴₂𝐾_𝐺^𝐻𝑈₂𝑊𝑉^𝐻₂𝐾_𝐽𝐵^𝐻₂𝐵^𝐻₁^†=𝐴^†₁𝐴₂𝐾_𝐺^𝐻𝐹𝐾_𝐽𝐵^𝐻₂𝐵^𝐻₁^†,𝒫_{(𝐴2𝐾𝐺𝐻)}^𝐻𝑈₂𝑊𝑉^𝐻₂𝒫_𝐽−𝒫_𝐺^𝐻𝒫_{(𝐴2𝐾𝐺𝐻)}^𝐻𝑈₂𝑊𝑉^𝐻₂𝑃_𝐽𝒫_𝐵^𝐻₂=𝒫_{(𝐴2𝐾𝐺𝐻)}^𝐻𝐹𝒫_𝐽−𝒫_𝐺^𝐻𝒫_{(𝐴2𝐾𝐺𝐻)}^𝐻𝐹𝑃_𝐽𝒫_𝐵^𝐻₂. (45) Substituting the expressions above into (33) yields that 𝑋=𝐴^†₁𝐸𝐵^𝐻₁^†+𝑓𝐴₁,𝐴₂,𝐵₁,𝐵₂,𝐸−𝐴^†₁𝐴₂𝑓𝐴₁,𝐴₂,𝐵₁,𝐵₂,𝐸𝐵^𝐻₂𝐵^𝐻₁^†−𝑃+𝐹_𝐴𝐻1+𝒫_𝐺^𝐻𝐹𝒫_𝐵^𝐻₁+𝒫_𝐵^𝐻₂−𝐴^†₁𝐴₂𝐾_𝐺^𝐻𝐹𝐾_𝐽𝐵^𝐻₂𝐵^𝐻₁^†−𝒫_{(𝐴2𝐾𝐺𝐻)}^𝐻𝐹𝒫_𝐽+𝒫_𝐺^𝐻𝒫_{(𝐴2𝐾𝐺𝐻)}^𝐻𝐹𝑃_𝐽𝒫_𝐵^𝐻₂. (46)

We have the following theorem.

Theorem 3. Let 𝐴₁=𝐴𝑆,𝐴₂=𝐴𝐼_𝑛,𝐵−𝑆₁=𝐵𝑅,𝐵₂=𝐵𝐼_𝑛−𝑅. (47) The matrix equation (2) with constraint 𝑆𝑋=𝑋𝑅 is consistent if and only if 𝒫_𝐺𝐾_𝐴1𝐸𝒫_𝐵2=𝐾_𝐴1𝐸,𝒫_𝐴2𝐸𝐾_𝐵1𝒫_𝐽^𝐻=𝐸𝐾_𝐵1, (48) with 𝐺=𝐾_𝐴1𝐴₂,𝐽=𝐵^𝐻₂𝐾_𝐵1. (49) In the meantime, a general solution is given by 𝑋=𝐴^†₁𝐸𝐵^𝐻₁^†+𝑓𝐴₁,𝐴₂,𝐵₁,𝐵₂,𝐸−𝐴^†₁𝐴₂𝑓𝐴₁,𝐴₂,𝐵₁,𝐵₂,𝐸𝐵^𝐻₂𝐵^𝐻₁^†−𝑃+𝐹_𝐴𝐻1+𝒫_𝐺^𝐻𝐹𝒫_𝐵^𝐻₁+𝒫_𝐵^𝐻₂−𝐴^†₁𝐴₂𝐾_𝐺^𝐻𝐹𝐾_𝐽𝐵^𝐻₂𝐵^𝐻₁^†−𝒫_{(𝐴2𝐾𝐺𝐻)}^𝐻𝐹𝒫_𝐽+𝒫_𝐺^𝐻𝒫_{(𝐴2𝐾𝐺𝐻)}^𝐻𝐹𝑃_𝐽𝒫_𝐵^𝐻₂, (50) where the arbitrary matrix 𝐹 satisfies 𝑆𝐹=𝐹𝑅 and 𝑓(𝐴₁,𝐴₂,𝐵₁,𝐵₂,𝐸) is determined by (36).

4. Eigenvector-Free Formulas of the General Solutions to (2) with 𝑃𝑋=𝑠𝑋𝑄 Constraint

For this constraint, if we set 𝑆 and 𝑅 as (4), it is not difficult to verify that 𝑆, 𝑅 are Hermitian idempotent, and the constraint 𝑃𝑋=𝑠𝑋𝑄 is equivalent to 𝑆𝑋=𝑋𝑅. (51) By Theorem 3, we have the following theorem.

Theorem 4. Let 𝐴₁=12𝐴𝐼_𝑛+𝑃,𝐴₂=12𝐴𝐼_𝑛,𝐵−𝑃₁=12𝐵𝐼_𝑛+𝑠𝑄,𝐵₂=12𝐵𝐼_𝑛−𝑠𝑄. (52) The matrix equation (2) with constraint 𝑃𝑋=𝑠𝑋𝑄 is consistent if and only if 𝒫_𝐺𝐾_𝐴1𝐸𝒫_𝐵2=𝐾_𝐴1𝐸,𝒫_𝐴2𝐸𝐾_𝐵1𝒫_𝐽^𝐻=𝐸𝐾_𝐵1, (53) with 𝐺=𝐾_𝐴1𝐴₂,𝐽=𝐵^𝐻₂𝐾_𝐵1. (54) In the meantime, a general solution is given by 𝑋=𝐴^†₁𝐸𝐵^𝐻₁^†+𝑓𝐴₁,𝐴₂,𝐵₁,𝐵₂,𝐸−𝐴^†₁𝐴₂𝑓𝐴₁,𝐴₂,𝐵₁,𝐵₂,𝐸𝐵^𝐻₂𝐵^𝐻₁^†−𝑃+𝐹_𝐴𝐻1+𝒫_𝐺^𝐻𝐹𝒫_𝐵^𝐻₁+𝒫_𝐵^𝐻₂−𝐴^†₁𝐴₂𝐾_𝐺^𝐻𝐹𝐾_𝐽𝐵^𝐻₂𝐵^𝐻₁^†−𝒫_{(𝐴2𝐾𝐺𝐻)}^𝐻𝐹𝒫_𝐽+𝒫_𝐺^𝐻𝒫_{(𝐴2𝐾𝐺𝐻)}^𝐻𝐹𝑃_𝐽𝒫_𝐵^𝐻₂, (55) where the arbitrary matrix 𝐹 satisfies 𝑃𝐹=𝑠𝐹𝑄 and 𝑓(𝐴₁,𝐴₂,𝐵₁,𝐵₂,𝐸) is determined by (36).

5. Numerical Examples

In this section, we present some numerical examples to illustrate the effectiveness of Theorems 3 and 4. For simplicity, we set 𝑚=𝑛=𝑝 and restrict the coefficient matrices 𝐴, 𝐵 and the right-hand-sided matrix 𝐸 to ℛ^𝑛×𝑛. The coefficient matrices 𝐴, 𝐵 are randomly constructed by 𝐴=𝑈diag𝜎₁,…,𝜎_𝑛𝑉^𝑇, (56) where the orthogonal matrices 𝑈 and 𝑉 are constructed as follows: 𝑈,temp=qr(1−2rand(𝑛)),𝑉,temp=qr(1−2rand(𝑛)), (57) and the singular values {𝜎_𝑖} will be chosen at interval (0,1). For the computational value 𝑋 of (2) with constraint 𝑃𝑋=𝑠𝑋𝑄 or 𝑆𝑋=𝑋𝑅, the residual error 𝜖_𝑋, the 𝑃𝑄-commuting error 𝜖_𝑃𝑄, 𝑆𝑅-commuting error 𝜖_𝑆𝑅, and consistent error Cond_err are denoted by 𝜖_𝑋=𝐸−𝐴𝑋𝐵^𝐻_𝐹,𝜖_𝑃𝑄=‖𝑃𝑋−𝑠𝑋𝑄‖_𝐹,𝜖_𝑆𝑅=‖𝑆𝑋−𝑋𝑅‖_𝐹,Cond_err=max𝒫_𝐺𝐾_𝐴1𝐸𝒫_𝐵2−𝐾_𝐴1𝐸_𝐹,𝒫_𝐴2𝐸𝐾_𝐵1𝒫_𝐽^𝐻−𝐸𝐾_𝐵1𝐹. (58)

Example 1. In this example, we test the solutions to (2) with 𝑆𝑋=𝑋𝑄 constraint by Theorem 3. The coefficient matrices 𝐴, 𝐵 are constructed as in (56), and the right-hand-sided matrix 𝐸 is constructed as follows: 𝐸=𝐴𝑋_∗𝐵^𝐻, (59) where 𝑋_∗ satisfies 𝑅𝑋_∗=𝑋_∗𝑆, (60) and 𝑆, 𝑅 are symmetric idempotent. That implies that the constrained equation (2) is consistent, so the residual error 𝜖_𝑋 and consistent error Cond_err should be zero with the computational value 𝑋.
For different 𝑛, the residual error 𝜖_𝑋, 𝑆𝑅-commuting error 𝜖_𝑆𝑅, and consistent errors Cond_err can reach the precision 10⁻⁰⁹, but all of them seem not to depend on the matrix size 𝑛 very much, and the CPU time also grows quickly as 𝑛 increases. In Table 1, we list the CPU time, 𝜖_𝑋, 𝜖_𝑆𝑅, and Cond_err, respectively.

Table 1 

Variant matrix sizes 𝑛 for the solutions to (2) with 𝑆𝑋=𝑋𝑅 constraint.

Example 2. We test the solutions to (2) with 𝑃𝑋=𝑋𝑄 constraint by Theorem 4. The test matrices 𝐴, 𝐵, and 𝐸 are constructed as in (56) with 𝑋_∗ satisfying 𝐸=𝐴𝑋_∗𝐵^𝐻, (61) where 𝑋_∗ satisfies 𝑃𝑋_∗=𝑋_∗𝑄, (62) and 𝑃, 𝑄 are symmetric involutory.
For different 𝑛, the numerical result is similar to those of Example 1; that is, the residual error 𝜖_𝑋, 𝑃𝑄-commuting error 𝜖_𝑃𝑄, and consistent errors Cond_err can all reach the precision 10⁻⁰⁹, but it seems that they do not depend on the matrix size 𝑛 very much. However, the CPU time grows quickly as 𝑛 increases. In Table 2, we list the CPU time, 𝜖_𝑋, 𝜖_𝑃𝑄, and Cond_err, respectively.

Table 2 

Variant matrix sizes 𝑛 for solutions to (2) with 𝑃𝑋=𝑋𝑄 constraint.

6. Conclusion

In this paper, we consider (2) with two special constraints 𝑃𝑋=𝑠𝑋𝑄 and 𝑆𝑋=𝑋𝑅, where 𝑃, 𝑄∈𝒞^𝑛×𝑛 are Hermitian involutory, 𝑆, 𝑅∈𝒞^𝑛×𝑛 are Hermitian idempotent, and 𝑠=±1. We represent the general solutions to the constrained equation by eigenvalue decompositions of 𝑃, 𝑄, 𝑆, 𝑅, release the involved eigenvector by Moore-Penrose generalized inverses, and get the eigenvector-free formulas of the general solutions.

Acknowledgments

The author is grateful to the referees for their enlightening suggestions. Moreover, the research was supported in part by the Natural Science Foundation of Zhejiang Province and National Natural Science Foundation of China (Grant nos. Y6110639, LQ12A01017, and 11201422).