The Least Squares Hermitian (Anti)reflexive Solution with the Least Norm to Matrix Equation

Abstract

For a given generalized reflection matrix , that is, , , where is the conjugate transpose matrix of , a matrix is called a Hermitian (anti)reflexive matrix with respect to if and By using the Kronecker product, we derive the explicit expression of least squares Hermitian (anti)reflexive solution with the least norm to matrix equation over complex field.

1. Introduction

Throughout this paper, we denote the set of all matrices over the real field and complex field by and , respectively. Let the symbols , , , , and stand for the set of all complex Hermitian matrices, the set of all real symmetric matrices, the set of all real skew symmetric matrices, the identity matrix, and the zero matrix with appropriate size, respectively. We also denote the transpose, the conjugate transpose, the Moore-Penrose inverse, and the Frobenius norm of by , , , and , respectively. Let ; we define , where is the -column of ,  .

In this paper, we consider the Hermitian (anti)reflexive solution problem of matrix equation Let us first recall the definition of those two kinds of matrices.

Definition 1. Given a generalized reflection matrix , that is, , , a matrix is called a Hermitian reflexive matrix with respect to if and The set of all Hermitian reflexive matrices with respect to is denoted by

Definition 2. Given a generalized reflection matrix , that is, , , a matrix is called a Hermitian antireflexive matrix with respect to if and The set of all Hermitian antireflexive matrices with respect to is denoted by

As an important kind of matrices, the reflexive matrices with different constraints were studied by many authors in recent years in literatures and references therein [1–29]. For instance, Zhao et al. [1] derived the general Hermitian reflexive solution to , while Yu and Shen [2] considered a more general case, that is, the Hermitian -(anti)-reflexive solution to . As an extension of matrix equation , the matrix system was considered by Zhou and Yang [3]. They gave the necessary and sufficient conditions for the existence of a Hermitian reflexive solution to the above matrix system. As we can see, many important associated works were widely studied. But finding the least squares Hermitian (anti)reflexive solution with the least norm to matrix equation is still a problem. Therefore, we in this paper will fill this gap.

We consider the following two problems.

Problem 3. Given matrices ,  , and , let Find such that

Problem 4. Given matrices ,  , and , let Find such that

Now, we start with some useful and fundamental results in the following.

Lemma 5. The least squares solution to the matrix equation with and is given by , where is an arbitrary vector. And the least squares solution with the least norm is

Kronecker product method plays an important role in this paper. We use this method to get some important vectorizing formulas.

For any , , where , The real representation matrix of is given by , which has the following properties:

(1)

(2)

(3) For , we define Then

Analogously to the method of Yuan et al. (see [30]), we can get a fundamental result as below.

Lemma 6. Let , , and be complex matrices with appropriate sizes; then

Proof. We denote , , and Then By the rule of we have

The structure of Hermitian (anti)reflexive matrix promises the following decompositions, which would simplify our problems.

Lemma 7. Let be the generalized reflection matrix. Then there exists an unitary matrix such that

Lemma 8 (see [18]). (1) Let and let the spectral decomposition of the reflection matrix be given as (7). Then if and only ifwhere and
(2) See [4]. Let and let the spectral decomposition of the reflection matrix be given as (7). Then if and only ifwhere

2. The Solution of Problem 3

In this section, we provide the least squares Hermitian reflexive solution with the least norm to matrix equation . There are the vectorizing formulas of Hermitian matrix, which are very useful in what comes below.

Definition 9. Let ; we definewhere , ,  , and

Definition 10. Let ; we definewhere , , and

Lemma 11 (see [30]). Suppose that Then if and only if , where is given by (10), whereObviously, is the -column of , , and .

Lemma 12. Suppose that Then if and only if , where is given by (11), whereObviously, is the -column of , , and .

Let , , Then and By Lemmas 11 and 12, one has

If , then, by Lemma 8, there exists a unitary matrix such thatwhere , and

Set where is given by (7). , are given by (12) with the sizes , , respectively. , are given by (13) with the sizes , , respectively. Then one has the following result.

Theorem 13. Let , , and . Then the solution to matrix equation is given by where is given by (7), , and is an arbitrary vector. And the solution with the least norm is given by

Proof. Since it follows from (14) thatNow, by Lemmas 6 and 11 and (23), we have where is given by (17); . Then, by Lemma 5, is the required least squares solution, where is an arbitrary vector. Recall that Therefore Then the least squares Hermitian reflexive solution is given by (18). Obviously, the least squares Hermitian reflexive solution with the least norm is given by (20).

3. The Solution of Problem 4

In this section, we provide the least squares Hermitian antireflexive solution with the least norm to matrix equation

We first derive the relation between and .

Lemma 14 (see [31]). Let be any matrix. Then wherewith , , , being an matrix with the element at position being and the others being . Moreover, is an orthogonal matrix, which is uniquely determined by the integers

Let ; then, by Lemma 8, there exists a unitary matrix such that where and is given by (7). Since ,

By Lemma 14, one hasSet and is given by (29). Then one has the following results.

Theorem 15. Let , , and Then on has the following:
(1) The solution to matrix equation is given bywhere is given by (7), , and is an arbitrary vector.
(2) The solution with the least norm is given bywhere is given by (7) and , .

Proof. It follows from Lemma 6 that Combining (32) and (37), we obtain Then where and
By Lemma 5, the least squares solution is given bywhere is an arbitrary vector.
The least squares solution with least norm is
So Now, it is easy to see that our required solutions are given by (34) and (36).

4. Conclusion

By exploiting the decomposition of Hermitian (anti)reflexive matrix and the Kronecker product, we derive the expression of the least squares Hermitian (anti)reflexive solution with the least norm to This method overcomes the difficulty of finding the structured least squares solution with the least norm but creates a large-scale linear system Therefore, it fits well with the small-scale coefficient matrix, but, for the large-scale case, it may not work so well.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Both authors read and approved the final manuscript.

Acknowledgments

This research was supported by Macau Science and Technology Development Fund (no. 003/2015/A1) and a grant from the National Natural Science Foundation of China (11571220).