Research Article | Open Access
Eigenvector-Free Solutions to the Matrix Equation with Two Special Constraints
The matrix equation with or constraint is considered, where S, R are Hermitian idempotent, P, Q are Hermitian involutory, and . By the eigenvalue decompositions of S, R, the equation with 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 with constraint.
In , Chen has denoted a square matrix , the reflexive or antireflexive matrix with respect to by where the matrix 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 with constraint where the matrices , , , the Hermitian involutory matrices , the Hermitian idempotent matrices , and the scalars .
Equation (2) with different constraints such as symmetry, skew-symmetry, and , 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 ; the authors achieved symmetric, skew-symmetric, and positive semidefinite solutions to (2) by quotient singular value decomposition (QSVD) [24, 25]. Moreover, in , CCD  was used for establishing a formula of the general solutions to (2) with diagonal constraint.
In , we have presented an eigenvector-free solution to the matrix equation (2) with constraint , where we represented its general solution and existence condition by -inverses of the matrices , , and . Note that the -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 or another constraint ; our ideas are based on the following observations. (1)If we set then and are both Hermitian idempotent. The above fact implies is the special case of . So, we only discuss (2) with constraint and construct the constrained solution by selecting suitable matrices , as (4). (2)With the eigenvalue decompositions (EVDs) of the Hermitian matrices , , matrix with constraint can be rewritten in (lower dimensional) two free variables and . And the corresponding constrained problem can be equivalently transformed to an unconstrained equation with given coefficient matrices , , (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 , , [15, 20, 27–29]. However, the formulas above are maybe not simpler because the coefficient matrices contain the eigenvectors of , . In fact, the Hermitian idempotence of the matrices , 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 , 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 constraint by the EVDs of , . In Section 3, we present the corresponding eigenvector-free representations. Equation (2) with constraint is regarded as the special case of (2) with 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 denote the space of complex matrix. For a matrix , and denote its transpose and Moore-Penrose generalized inverse, respectively. Matrix is identity matrix with order ; refers to zero matrix, and is the zero matrix with order . For any matrix , we also denote So,
2. Solution to (2) with Constraint by the EVDs
For the Hermitian idempotent matrices , , let be their two eigenvalue decompositions with unitary matrices , , respectively. Then holds if and only if where . And the constrained solution can be expressed in Partitioning , and using the transformations (10), (2) with constraint is equivalent to the following unconstrained problem: where
For the unconstrained problem (11), we introduce the results about its existence conditions and expression of solutions.
Lemma 1. Given , , , , and , the linear matrix equation is consistent if and only if or, equivalently, if and only if where and . And a representation of the general solution is with where the matrices and are arbitrary.
The lemma is easy to verify; we can turn to  for details. The difference between them is that we replace the -inverse in the theorem of  by the corresponding Moore-Penrose generalized inverse, and the expression of solutions is complicated relatively. However, compared with the multiformity of the -inverses, the Moore-Penrose generalized inverse involved representation is unique and fixed.
Theorem 2. The matrix equation with constraint is consistent if and only if where In the meantime, a general solution is given by where the matrices and are arbitrary.
3. Eigenvector-Free Formulas of the General Solutions to (2) with Constraint
The existence conditions and the expression of the general solution given in Theorem 2 contain the eigenvector matrices of , , 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 Note that is the Moore-Penrose generalized inverse of , which gives where Then Set and denote It is not difficult to verify that together with Then the first equality of (17) can be rewritten as and the other can be rewritten as Now, we consider the simplification of the general solution given by (10), which can be rewritten as Note that Together with (26), so we can represent by a given expression of , , . Let Hence, we have Since then Letting it is not difficult for us to verify . Together with the following equality holds: Note that Then Hence, Substituting the expressions above into (33) yields that
We have the following theorem.
Theorem 3. Let The matrix equation (2) with constraint is consistent if and only if with In the meantime, a general solution is given by where the arbitrary matrix satisfies and is determined by (36).
4. Eigenvector-Free Formulas of the General Solutions to (2) with Constraint
Theorem 4. Let The matrix equation (2) with constraint is consistent if and only if with In the meantime, a general solution is given by where the arbitrary matrix satisfies and 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 and restrict the coefficient matrices , and the right-hand-sided matrix to . The coefficient matrices , are randomly constructed by where the orthogonal matrices and are constructed as follows: and the singular values will be chosen at interval . For the computational value of (2) with constraint or , the residual error , the -commuting error , -commuting error , and consistent error are denoted by
Example 1. In this example, we test the solutions to (2) with constraint by Theorem 3. The coefficient matrices , are constructed as in (56), and the right-hand-sided matrix is constructed as follows:
and , are symmetric idempotent. That implies that the constrained equation (2) is consistent, so the residual error and consistent error should be zero with the computational value .
For different , the residual error , -commuting error , and consistent errors can reach the precision , but all of them seem not to depend on the matrix size very much, and the CPU time also grows quickly as increases. In Table 1, we list the CPU time, , , and , respectively.
Example 2. We test the solutions to (2) with constraint by Theorem 4. The test matrices , , and are constructed as in (56) with satisfying
and , are symmetric involutory.
For different , the numerical result is similar to those of Example 1; that is, the residual error , -commuting error , and consistent errors can all reach the precision , but it seems that they do not depend on the matrix size very much. However, the CPU time grows quickly as increases. In Table 2, we list the CPU time, , , and , respectively.
In this paper, we consider (2) with two special constraints and , where , are Hermitian involutory, , are Hermitian idempotent, and . We represent the general solutions to the constrained equation by eigenvalue decompositions of , , , , release the involved eigenvector by Moore-Penrose generalized inverses, and get the eigenvector-free formulas of the general solutions.
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).
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