A General Hermitian Nonnegative-Definite Solution to the Matrix Equation AXB = C

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1. Introduction

Let represent a matrix in the vector space of complex (real) matrices, and let denote the conju- gate transpose (transpose) of. We frequently encounter linear matrix equations of the form

(1)

with, , and. Using the Moore-Penrose inverse, Penrose in [2] was the first to provide conditions for the existence and represen- tation of the general solution to (1). Since then, numerous authors have derived representations of a general solution to (1) under varying restrictions on, , and and on the type of solution. Existence conditions and alternative expressions for the general solution have been studied by Dogaru in [3] and Chu in [4] . Also, Rosen in [5] has provided a representation of the general solution for (1) when.

Hermitian solutions to (1) have been considered by numerous authors as well such as by Khatri in [6] , Wang, Yan, and Dai in [7] , and Cvetković-Ilić in [8] . Additionally, Wang and Yang in [9] and Cvetković-Ilić and Dragana in [10] have found necessary and sufficient conditions for the existence of a real nonnegative-definite (Re-n.n.d.) solution and a representation of a general Re-n.n.d. solution to (1). Also, Zhang has proposed representations of the general Hermitian n.n.d. solutions to (1) in [1] .

In this paper, we derive necessary and sufficient conditions for the existence of a Hermitian n.n.d. solution and a new representation of the general Hermi- tian n.n.d. solution to (1). Moreover, our representation is invariant with respect to the generalized inverse (g-inverse) involved, unlike the solution from Khatri in [6] . We then apply our solution to an example problem posed by Zhang in [1] and obtain a simpler solution that contradicts the proposed solution from Zhang in [1] . Furthermore, while Zhang employs an algorithmic method in [1] , we obtain a closed-form solution. We also provide an example application where we employ our general Hermitian n.n.d. solution to demonstrate that two matrix quadratic forms are stochastically independent.

2. Notation and Definitions

In this section, we establish some notation to be used throughout the remainder of the paper. We use to represent the identity matrix and use to denote the identity matrix if the order of the matrix is apparent. We use to denote the column space (range space) and to denote the row space of. The rank of is represented by. We let

denote the cone of all Hermitian (symmetric) n.n.d. matrices in, where is the set of all complex (real) matrices.

Given a matrix, a g-inverse of is a matrix that satisfies the property. Finally, we let denote the set of complex Hermitian matrices.

3. Mathematical Preliminaries

This section contains the fundamental mathematical results that will be used in this paper. We provide a definition of parallel summable matrices and introduce five lemmas that are essential to our main results.

Definition. Let. A pair of matrices is defined to be parallel summable if is invariant under the choice of the g-inverse. That is, if

or, equivalently,

then the parallel sum of and is

We provide useful results for parallel summable matrices that are included in the next two lemmas. The first two lemmas are from Rao in [11] .

Lemma 3.1. ( [11] , Lemma 2.2.4) Let, , and. If and, then is invariant to the choice of the g-inverse.

Lemma 3.2. ( [11] , Theorem 10.1.8) For a pair of parallel summable matrices, we have.

The following lemma comes from Khatri and Mitra in [12] and is used in the proof of the main result of this paper.

Lemma 3.3. Let, , and such that is consistent. Then, is a representation of a general solution for

(2)

if and only if is a representation of the general solution for

(3)

The following lemma verifies that, under certain conditions, a quadratic form is invariant under the choice of the g-inverse. Moreover, we verify that the quadratic form is n.n.d.

Lemma 3.4. Let and. Also, let

(4)

with

(5)

(6)

(7)

and

(8)

If, then and is invariant to the choice of.

Proof. First, because, there exists a. Also, we have that and. By Lemma 4.2.2 and Theorem 4.4.6 from Harville in [13] , we have and. Also, from Lemma 4.5.10 of Harville in [13] , we see that and. Thus, by Lemma 3.1, the lemma holds.

The following lemma can be found in Theorem 1 from Albert in [14] .

Lemma 3.5. Let, where, , and. Then, if and only if, , and.

We use the following lemma in the proof of the second example. The lemma is well-known, and, therefore, is stated without proof.

Lemma 3.6. If is a n.n.d. matrix and and are matrices such that, then is equivalent to.

4. A General Hermitian N.N.D. Solution to AXB = C

In [6] , Khatri provided existence conditions and have proposed a representation of the Hermitian n.n.d. solution to

(9)

where and. However, as noted by Baksalary in [15] , his results are dependent on the choice of the g-inverse and, hence, do not represent a general Hermitian n.n.d. solution to (9).

In their efforts to derive a solution, Khatri and Mitra in [12] have employed an innovative technique that converts (9) to an equation in which the coefficient matrices are equal. We call this technique “symmetrization” because it effectively transforms (9) from a matrix bilinear form in and to the matrix equation form

where. We employ this symmetrization device in the proof of our main result.

The following theorem provides necessary and sufficient conditions for the existence of and a representation of the general Hermitian n.n.d. solution to (9) that is invariant to the choice of g-inverse. We remark that the general Hermitian n.n.d. solution given below in (11) is based on a result following Theorem 1 of Groß in [16] .

Theorem. Let and such that (9) is consistent. Then, (9) has a Hermitian n.n.d. solution if and only if is defined as in (4) and

(10)

A representation of the general Hermitian n.n.d. solution is

(11)

where represents the class of g-inverses of given by

(12)

such that are arbitrary solutions of

(13)

and

(14)

respectively. Also, , is arbi- trary but fixed, and and are free to vary. We remark that the form of the specialized g-inverse in (12) comes from Theorem 1 of Groß in [16] .

Proof. First, assume is a solution to (9). Then,

so that. Next, let, where with. Then, using Lemma 3.2, we have

Similarly, we have that.

Next, assume (4) and (10) hold. Following Khatri and Mitra in [6] , we first write (13) and (14) as and, respectively, where, and are defined in (5)-(8), respectively. One can check that and. From Lemma 3.1, we have that and are invariant with respect to. Thus, by Theorem 2.2 of Khatri and Mitra in [6] and the fact that there exists a, general Hermitian n.n.d. solutions to (13) and (14) are

(15)

and

(16)

respectively, where are arbitrary. Also, because and, we have and, and, hence,

(17)

and

(18)

Using Equations (15) and (16), we have that

(19)

Adding to the right-hand side of (19) and letting and, we have that

(20)

where and

By Lemma 3.4,. Also, using (17) and (18), we get that by Lemma 3.5. Thus, the right-hand side of equation (20) is Hermitian n.n.d., and, therefore,. Because , we have from Theorem 1 of Groß (2000) that

(21)

has a Hermitian n.n.d. solution.

Next, let be given by (11). Then,

Thus, if (21) has a Hermitian n.n.d. solution, then (9) has a Hermitian n.n.d. solution and, moreover, every Hermitian n.n.d. solution to (21) is a Hermitian n.n.d. solution to (9).

Now, let be a solution to (9). Also, let, , and recall that. Then, is a solution to (21). Thus, (11) is a general Hermitian n.n.d. solution to (9).

In our theorem, we derived a general Hermitian n.n.d. solution to (9) for the case where. We next present the main result of the paper. We consider the general case by relaxing the n.n.d. and equal dimension constraints on the coefficient matrices and.

Corollary 1. Let, , and such that

(22)

is consistent. Then, (22) has a Hermitian n.n.d. solution if and only if

and

where. A representation of the general Hermitian n.n.d. solution to (22) is given by

(23)

such that represents the class of g-inverses of given by

(24)

where are arbitrary solutions of

(25)

and

(26)

respectively, such that, and are free to vary, and is arbitrary but fixed.

Proof. The corollary follows from Lemma 3.3 and the theorem.

5. Two Examples

We now provide two example applications of our main results in Section 4, which were performed using R version 3.2.4.

5.1. Example 1

We utilize an example from Zhang in [1] to illustrate the computational ease and accuracy of our solution. Let

(27)

so that, , and. The goal is to determine all Hermitian n.n.d. solutions to

(28)

where, , and are given in (27).

We first give the general Hermitian n.n.d. solution from Zhang in [1] , which is of the form

(29)

where

and, and are parameters satisfying, , and, where with

Next, we present our general Hermitian n.n.d. solution to (28). Using Corollary 4.1, we have

where. Therefore, a Hermitian n.n.d. solution to (28) exists. Note that

where. Next, we employ (25) and (26) to obtain

and

We remark that in (23), and, thus, from (23), we have that

(30)

is the unique solution to (28) because.

The solution given in (30) contradicts the general Hermitian n.n.d. solution given in (29). We remark that our general solution is closed form and is not obtained algorithmically as that from Zhang in [1] .

5.2. Example 2

Next, consider the random matrix, where. Several authors have studied the independence of matrix normal-based quadratic forms, ,. Numerous results can be found in work by Mathai and Provost in [17] and Gupta and Nagar in [18] .

In the following corollary, we derive a representation of the general covari- ance structure of the form of a normal random matrix such that the two matrix quadratic forms and are independent when the coefficient matrices.

Corollary 2. Let with, , , and. Then, the two quadratic forms and are stochastically independent if and only if

(31)

where represents the class of generalized inverses defined by (12), is free to vary, and are arbitrary solutions of

and

Proof. By Theorem 6.6b.1 from Mathai and Provost in [17] , and are stochastically independent if and only if, , , and. However, a direct application of Lemma 3.6 reduces these conditions to the single equation. Thus, by the theorem in Section 4, and are stochastically independent if and only if (31) holds.

6. Discussion

In this paper, we derive necessary and sufficient conditions for the existence of a Hermitian n.n.d. solution and a new general Hermitian n.n.d. solution to the matrix equation. Unlike the proposed n.n.d. solution by Khatri and Mitra in [12] , our general representation of is invariant with respect to the choice of g-inverse. Moreover, using an example from Zhang in [1] , we demon- strate that our closed-form general Hermitian n.n.d. solution contradicts the proposed general Hermitian n.n.d. solution from Zhang in [1] . Finally, we apply our main result to obtain the general form of a matrix-normal random matrix with covariance matrix such that two matrix quadratic forms are independent.

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