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

We derive necessary and sufficient conditions for the existence of a Hermitian nonnegative-definite solution to the matrix equation AXB C = . Moreover, we derive a representation of a general Hermitian nonnegative-definite solution. We then apply our solution to two examples, including a comparison of our solution to a proposed solution by Zhang in [1] using an example problem given from [1]. Our solution demonstrates that the proposed general solution from Zhang in [1] is incorrect. We also give a second example in which we derive the general covariance structure so that two matrix quadratic forms are independent.


Introduction
in [4].Also, Rosen in [5] has provided a representation of the general solution for (1) when 0 C = .
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 Hermitian 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.

Notation and Definitions
In this section, we establish some notation to be used throughout the remainder of the paper.We use n I to represent the n n × identity matrix and use I to denote the identity matrix if the order of the matrix is apparent.We use ( ) to denote the column space (range space) and   is the set of all n n × complex (real) matrices.
Given a matrix ( ) Finally, we let H n  denote the set of complex Hermitian n n × matrices.

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 , 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.

L B A B C
. Also, from Lemma 4.5.10 of Harville in [13], we see that ( ) ( ) . Thus, by Lemma 3.1, the lemma holds.
The following lemma can be found in Theorem 1 from Albert in [14].
, where 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. where 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 A and B to the matrix equation form where n D ≥ ∈  .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]. 9)is consistent.Then, (9)   has a Hermitian n.n.d.solution if and only if T is defined as in ( 4) and A representation of the general Hermitian n.n.d.solution is where ( ) represents the class of g-inverses of ( ) , and respectively.Also, ( ∈  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 0 n X ≥ ∈  is a solution to (9).Then, , where Then, using Lemma 3.2, we have  are invariant with respect to T − .Thus, by Theorem 2.2 of Khatri and Mitra in [6] and the fact that there exists a n T − ≥ ∈  , general Hermitian n.n.d.solutions to (13) and ( 14) are

T B A B AG B A B AG G A A B B G B A B A BX B A B
and respectively, where 1 2 , n V V ≥ ∈  are arbitrary.Also, because ( ) ( ) ( ) ( ) Using Equations ( 15) and ( 16), we have that to the right-hand side of (19) and letting 1 V I = and 2 V P P * = , we have that has a Hermitian n.n.d.solution.

AX B A A B C C Y Z A B B A I A B A B U I A B A B B A A B C C Y Z A B B A A B C A B B B A B C A B A A A B C A B A B A B C A B B A B A B C A B A B C
( ) represents the class of g-inverses of H given by ( ) where and are free to vary, and ( ) Proof.The corollary follows from Lemma 3.3 and the theorem.

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

Example 1
We utilize an example from Zhang in [1] to illustrate the computational ease and accuracy of our solution.Let so that 2 n = , 4 m = , and 3 p = .The goal is to determine all Hermitian n.n.d.We first give the general Hermitian n.n.d.solution from Zhang in [1], which is of the form ( ) and , , a b c , and d are parameters satisfying 2.4448 a > , 1 0 b > , and , where We remark that ( ) 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].

Example 2
Next, consider the random matrix ( ) 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 covariance structure of the form V = Ψ ⊗ Σ of a normal random matrix such that the two matrix quadratic forms X AX ′ and X BX ′ are independent when the coefficient matrices ′ Ψ Ψ = , and 0 M A BM ′ Ψ = .However, a direct application of Lemma 3.6   reduces these conditions to the single equation 0 A B Ψ =.Thus, by the theorem in Section 4, X AX ′ and X BX ′ are stochastically independent if and only if (31) holds.

Discussion
In this paper, we derive necessary and sufficient conditions for the existence of a Mitra in [12], our general representation of X is invariant with respect to the choice of g-inverse.Moreover, using an example from Zhang in [1], we demonstrate 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 V = Ψ ⊗ Σ such that two matrix quadratic forms are independent.


 denote the cone of all n n × Hermitian (symmetric) n.n.d.matrices in

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 (4) and(10) hold.Following Khatri and Mitra in[6], we first write (13) and (14) as YE F = and KZ L = , respectively, where , , E F K , and L are defined in (5)-(8), respectively.One can check that LK T E F 3.1, we have that FT F − * and L T L * − 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 (Then, 0 X 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 A and B . is consistent.Then, (22) has a Hermitian n.n.d.solution if and only if , of the general Hermitian n.n.d.solution to (22) is given by where A , B , and C are given in (27).
Theorem 6.6b.1 from Mathai and Provost in[17], Hermitian n.n.d.solution and a new general Hermitian n.n.d.solution to the matrix equation AXB C = .Unlike the proposed n.n.d.solution by Khatri and we present our general Hermitian n.n.d.solution to (28).Using