The common Re-nonnegative definite and Re-positive definite solutions to the matrix equations $A_1XA_1^\ast = C_1$ and $A_2XA_2^\ast = C_2$

1.   Introduction

For convenience, the notations used in this paper are given in Table 1.

A lot of papers have been published for solving the Re-nnd and Re-pd solutions to some matrix equations. For example, the Re-nnd and Re-pd solutions to $AX = C$ have been considered by Wu ^[1], Wu and Cain ^[2] and Groß ^[3]. The Re-nnd and Re-pd solutions to $AXB = C$ have been discussed by Wang and Yang ^[4], Cvetković-Iliíc ^[5], Tian ^[6] and Yuan and Zuo ^[7]. The common Re-nnd and Re-pd solutions to $AX = C$ and $XB = D$ have been investigated by Liu ^[8] and Yuan et al., ^[9]. Although there are some results ^[10,11,12] concerning the Hermitian nonnegative definite and positive definite solutions of the following matrix equations

where $A_1 \in {\mathbb{C}}^{m \times n}, A_2 \in {\mathbb{C}}^{p \times n}, C_1\in {\mathbb{C}}^{m \times m}$ and $C_2 \in {\mathbb{C}}^{p \times p}$, the research results on the Re-nnd and Re-pd solutions to the Eq (1) are quite limited so far. Recently, determinantal representations of solutions and Hermitian solutions to Eq (1) have been considered by Kyrchei ^[13]. Song and Yu ^[14] studied the common Re-nnd and Re-pd solutions of (1) firstly by utilizing the maximal and minimal inertias of the linear Hermitian matrix function and the generalized inverses of matrices, and established some necessary and sufficient conditions for the existence of Re-nnd and Re-pd solutions of Eq (1).

The purpose of this paper is to provide an alternative approach to solve Eq (1). The necessary and sufficient conditions for the solvability along with the expressions for the Re-nnd and Re-pd solutions to the Eq (1) are presented with the help of the Moore-Penrose inverses and the spectral decompositions of matrices.

2.   Some lemmas

Some lemmas are needed in the following.

Lemma 1. ^[15,16] Let $A_1 \in {\mathbb{C}}^{m \times n}, B_1 \in {\mathbb{C}}^{p \times q},$ $A_2\in {\mathbb{C}}^{l \times n}, B_2 \in {\mathbb{C}}^{p \times k}$ and $C_1 \in {\mathbb{C}}^{m \times q}, \; C_2 \in {\mathbb{C}}^{l \times k}.$ Then the pair of equations $A_1XB_1 = C_1, A_2XB_2 = C_2$ have a common solution $X$ if and only if

where $\mathcal{T} = \mathcal{R}(A_1^\ast)\cap \mathcal{R}(A_2^\ast), \ \mathcal{S} = \mathcal{R}(B_1)\cap \mathcal{R}(B_2).$ In this case, the general common solution to the equations can be expressed as

where

$A = \left[ \begin{array}{c} A_1 \\ A_2 \\ \end{array} \right], B = [B_1, B_2], \Psi = A_1^† A_1F_{A_2}, \Xi = E_{B_2}B_1B_1^†, D = A_2^† C_2B_2^† -A_1^† C_1B_1^†,$ and $V_1, V_2, V_3, V_4$ are arbitrary matrices.

Lemma 2. ^[17,18] Let $B \in {\mathbb{C}}^{n \times m}, C \in {\mathbb{C}}^{p \times n}$ and $A = A^\ast \in {\mathbb{C}}^{n \times n}.$ Then the matrix equation

has a solution $X \in {\mathbb{C}}^{m \times p}$ if and only if

In this case, all the solutions $X \in {\mathbb{C}}^{m \times p}$ satisfying Eq $(2)$ are given by

where $L = F_{C}BB^†$, $M\in {\mathbb{C}}^{m \times p}, S_X\in {\mathbb{C}}^{n \times n}$ are arbitrary matrices with $S_X^\ast = -S_X$ and $\Theta$ is given by $\Theta = \frac{1}{2}A(2I_n-BB^†)+\frac{1}{2}(\Phi-\Phi^\ast)BB^†$ with $\Phi = 2L^† F_{C}A+\left(I_n-L^† F_{C}\right)AL^† L$.

Lemma 3. ^[19,20] Let $A \in {\mathbb{C}}^{m \times n}$ and $D \in {\mathbb{C}}^{m \times n},$ and let the singular value decomposition of $A$ be $A = \tilde{U}\left[ \begin{matrix} \Sigma & 0\\ 0 & 0 \end{matrix} \right]\tilde{V}^\ast,$ where $\Sigma = \mathit{\mbox{diag}}(\sigma_{1}, ..., \sigma_{r}) > 0, r = \mathit{\mbox{rank}}(A)$, $\tilde{U} = [\tilde{U}_{1}, \tilde{U}_{2}]\in \mathbb{C}^{m\times m}, \tilde{V} = [\tilde{V}_{1}, \tilde{V}_{2}]\in \mathbb{C}^{n\times n}$ are unitary matrices with $\tilde{U}_{1}\in \mathbb{C}^{m\times r}, \tilde{V}_{1}\in \mathbb{C}^{n\times r}.$ Then:

$(a)$ The matrix equation $AY = D$ has a Hermitian nonnegative definite solution $Y \in {\mathbb{C}}^{n \times n}$ if and only if $DA^\ast\geq 0, \ \mathcal{R}(D) = \mathcal{R}(DA^\ast),$ in this case, the general Hermitian nonnegative definite solution is

where $Y_0 = A^† D+F_{A}(A^† D)^\ast+F_{A}D^\ast(DA^\ast)^† DF_{A},$ and $H\in {\mathbb{C}}^{n \times n}$ is an arbitrary Hermitian nonnegative definite matrix.

$(b)$ The matrix equation $AY = D$ has a Hermitian positive definite solution $Y \in {\mathbb{C}}^{n \times n}$ if and only if $AA^† D = D, \ \tilde{U}_1^\ast DA^\ast \tilde{U}_1 > 0,$ in this case, the general Hermitian positive definite solution is

where $Y_0 = A^† D+F_{A}(A^† D)^\ast+F_{A}D^\ast(DA^\ast)^{†}DF_{A},$ and $H\in {\mathbb{C}}^{n \times n}$ is an arbitrary Hermitian positive definite matrix.

3.   Re-nnd and Re-pd solutions to Eq (1)

In this section we propose a general theory elaborating how to solve the common Re-nnd and Re-pd solutions of Eq (1). From Lemma $1,$ Eq (1) has a common solution $X\in {\mathbb{C}}^{n \times n}$ if and only if

where $\mathcal{T} = \mathcal{R}(A_1^\ast)\cap \mathcal{R}(A_2^\ast).$ In this case, the general common solution of Eq (1) is

where $X_0$ is a particular common solution of Eq (1), which is given by

$A = \left[ \begin{array}{c} A_1 \\ A_2 \\ \end{array} \right], \Psi = A_1^† A_1F_{A_2}, D = A_2^† C_2(A_2^\ast)^† -A_1^† C_1(A_1^\ast)^†,$ and $V_1, V_2, V_3, V_4$ are arbitrary matrices. By Eq (5), we have

where $V_{12} = V_1+V_2^\ast, V_{34} = V_3+V_4^\ast$. Clearly, $X$ is Re-nnd (Re-pd) if and only if $H\geq 0$ ($H > 0$). By applying Lemma 2, we know that Eq (7) with respect to $V_{12}$ is solvable if and only if

In this case, the general solution is

where $M_{12}\in {\mathbb{C}}^{n \times n}$ and $S_{12}\in {\mathbb{C}}^{n \times n}$ are arbitrary matrices with $S_{12}^\ast = -S_{12}.$

Notice that

Therefore,

which implies that

Namely, $A^† AF_{A_1}$ and $A^† AF_{A_2}$ are the orthogonal projectors (see [21,p.80,Ex.63]):

from which it is easily deduced that

From Lemma 2, Eq (8) with respect to $V_{34}$ is solvable if and only if the following three equations hold simultaneously:

In the following, we will deduce the necessary and sufficient conditions on $H\geq 0$ ($H > 0$) such that Eqs (12)–(14) are solvable. Let

Then, by using the relations of (10) and (11) we get

By (15), the Eq (14) can be equivalently written as

Now, let the spectral decomposition of $A^† A$ be

where $a = \mbox{rank}(A^† A) = \mbox{rank}(A)$ and $U = [U_1, U_2]$ is a unitary matrix with $U_1\in {\mathbb{C}}^{n \times a}$. It follows from (11), (16), (17) and (19) that

where $P_1^2 = P_1 = P_1^\ast, P_2^2 = P_2 = P_2^\ast, P_0^2 = P_0 = P_0^\ast$ and $P_0P_1 = P_1, P_0P_2 = P_2.$ By applying the relations of (19) and (22), Eq (18) can be further simplified as

where

From Lemma $3,$ Eq (23) has a Hermitian nonnegative definite solution $H_{11} \in {\mathbb{C}}^{a \times a}$ if and only if

in this case, by simple calculations, we can obtain the general Hermitian nonnegative definite solution of Eq (23) is

where $D_{0} = U_1^\ast (X_0+X_0^\ast)U_1$ and

and $K\in {\mathbb{C}}^{a \times a}$ is an arbitrary Hermitian nonnegative definite matrix.

Suppose that the spectral decomposition of $I_a-P_0$ is

where $g = \mbox{rank}(I_a-P_0)$ and $W = [W_1, W_2]\in {\mathbb{C}}^{a \times a}$ is a unitary matrix with $W_1\in {\mathbb{C}}^{a \times g}$. From Lemma $3,$ Eq (23) has a Hermitian positive definite solution $H_{11} \in {\mathbb{C}}^{a \times a}$ if and only if

in this case, the general Hermitian positive definite solution is

where $H_{110}$ is given by (27) and $K\in {\mathbb{C}}^{a \times a}$ is an arbitrary Hermitian positive definite matrix.

By the ralations of (19)–(21) and (24), the Eqs (12) and (13) can be equivalently written as

Substituting (30) into (31) and (32), and noting that $P_0P_1 = P_1 = P_1P_0, P_0P_2 = P_2 = P_2P_0$, we obtain that

where

Direct verifications shows that $P_0-P_1$ and $P_0-P_2$ are orthogonal projectors. Hence, there exist unitary matrices $G\in \mathbb{C}^{a \times a}$ and $Q\in \mathbb{C}^{a \times a}$ such that

where $e = \mbox{rank}(P_0-P_1), f = \mbox{rank}(P_0-P_2)$, and $G_1 \in \mathbb{C}^{a \times e}, Q_1 \in \mathbb{C}^{a \times f}$ are column unitary matrices. By substituting the relations of (36) and (37) into (33) and (34), we arrive at the following equations:

Note that $G_1\in \mathbb{C}^{a \times e}$ and $Q_1\in \mathbb{C}^{a \times f}$ are column unitary matrices, it follows from ^[22] that the generalized singular value decomposition of the matrix pair $[G_1, Q_1]$ is of the following form:

where $M\in \mathbb{C}^{a \times a}$ is a nonsingular matrix and $E \in \mathbb{C}^{e \times e}, F \in \mathbb{C}^{f \times f}$ are unitary matrices, and

$k = \mbox{rank}\left([G_1, Q_1]\right) = e+f-s,$ and

with

Substituting (39) into (38) and Partitioning $M^\ast BM$ into the following form:

Then, by applying an established result in ^[9], we obtain

$(a)$ Eq (38) has a common Hermitian nonnegative definite solution $K$ if and only if

In this case, the general Hermitian nonnegative definite solution of $(38)$ can be expressed as

where

with

and $S \in {\mathbb{C}}^{k \times (a-k)}$ is an arbitrary matrix, $T\in {\mathbb{C}}^{(a-k) \times (a-k)}$ is an arbitrary Hermitian nonnegative definite matrix and $N\in {\mathbb{C}}^{(e-s) \times (k-e)}$ is an arbitrary contraction matrix $($i.e., the largest singular value of the matrix $N$ is not greater than $1)$.

$(b)$ Eq (38) has a common Hermitian positive definite solution $K$ if and only if

In this case, the general Hermitian positive definite solution of $(38)$ can be expressed as

where $F(K_{13})$ is defined by $(42)$ with

and $S \in {\mathbb{C}}^{k \times (a-k)}$ is an arbitrary matrix, $T\in {\mathbb{C}}^{(a-k) \times (a-k)}$ is an arbitrary Hermitian positive definite matrix and $N \in {\mathbb{C}}^{(e-s) \times (k-e)}$ is an arbitrary strict contraction matrix $($i.e., the largest singular value of the matrix $N$ is less than $1)$.

Once we achieve the Hermitian nonnegative definite (positive definite) solution $K$ of Eq (38), the matrix $H_{11}$ in (30) is completely specified. Also, when $H_{11}\geq 0$, by (24) and Theorem 1 of ^[23], we can determine the expression of the matrix $H\geq 0$ by

and when $H_{11} > 0$, by (24) and Theorem 1 of ^[23], we can determine the expression of the matrix $H > 0$ by

where $E, H_{12}, F$ and $R$ are arbitrary matrices with $F\geq 0$ and $R > 0.$

By using (19)–(21), the Eq (8) can be simplified as

where

When the conditions (12)–(14) hold, From Lemma 2, all the solutions $V_{11}^{(34)} \in {\mathbb{C}}^{a \times a}$ satisfying Eq $(48)$ are given by

where $L = P_1-P_2P_1$, $M_{34}\in {\mathbb{C}}^{a \times a}, S_{34}\in {\mathbb{C}}^{a \times a}$ are arbitrary matrices with $S_{34}^\ast = -S_{34}$ and $\Theta$ is given by

with

and $H$ being given by (47).

In summary of the discussion above, we have proved the following results.

Theorem 1. For given matrices $A_1 \in {\mathbb{C}}^{m \times n}, A_2 \in {\mathbb{C}}^{p \times n}, C_1\in {\mathbb{C}}^{m \times m}$ and $C_2 \in {\mathbb{C}}^{p \times p}$, let $A = \left[ \begin{array}{c} A_1 \\ A_2 \\ \end{array} \right], \Psi = A_1^† A_1F_{A_2}, D = A_2^† C_2(A_2^\ast)^† -A_1^† C_1(A_1^\ast)^†, \mathcal{T} = \mathcal{R}(A_1^\ast)\cap \mathcal{R}(A_2^\ast)$ and let $X_0, P_{\mathcal{T}_1}, P_{\mathcal{S}_1}$ and $P_\mathcal{L}$ be given by $(6)$, $(10)$ and $(15)$, respectively. Suppose that the spectral decompositions of $A^† A, I_a-P_0, P_0-P_1$ and $P_0-P_2$ are respectively given by $(19)$, $(28)$, $(36)$ and $(37)$ with $P_1, P_2$ and $P_0$ being given by $(20)$–$(22)$. Let $D_{0} = U_1^\ast (X_0+X_0^\ast)U_1$ and $B$ be given by $(35)$. Furthermore, assume that the generalized singular value decomposition of the matrix pair $[G_1, Q_1]$ is given by $(39)$, and the partition of the matrix $M^\ast BM$ is given by $(40).$ Then:

$(a)$ The Eq $(1)$ has a common Re-nnd solution if and only if

In this case, the general Re-nnd solution of $(1)$ can be expressed as

where

$V_{12}, V_{11}^{(34)}$ and $H_{11}$ are respectively given by $(9)$, $(49)$ and $(26)$ with $K$ being given by $(41)$, and $V_1, V_3, V_{12}^{(34)}, V_{21}^{(34)}, V_{22}^{(34)}, E$ and $F$ are arbitrary matrices with $F\geq 0$.

$(b)$ The Eq $(1)$ has a common Re-pd solution if and only if

In this case, the general Re-pd solution of $(1)$ can be expressed as

where

$V_{12}, V_{11}^{(34)}$ and $H_{11}$ are respectively given by $(9)$, $(49)$ and $(30)$ with $K$ being given by $(44)$, and $V_1, V_3, V_{12}^{(34)}, V_{21}^{(34)}, V_{22}^{(34)}, H_{12}$ and $R$ are arbitrary matrices with $R > 0$.

4.   Numerical algorithm and numerical examples

According to Theorem 1, we can describe a numerical algorithm to solve Eq $(1)$.

Algorithm 1.

$1).$ Input $A_{1}, A_{2}, C_{1}, C_{2}$.

$2).$ If the conditions $(4)$ are satisfied, go to $3$; otherwise, Eq $(1)$ has no solution, and stop.

$3).$ Compute $X_{0}$ by $(6)$.

$4).$ Compute $P_{\mathcal{T}_1}, P_{\mathcal{S}_1}$ and $P_{\mathcal{L}}$ by $(10)$ and $(15)$.

$5).$ Compute the spectral decomposition of the matrix $A^† A$ by $(19)$.

$6).$ Compute $P_{1}, P_{2}, P_{0}$ by $(20)-(22)$, respectively.

$7).$ Compute the spectral decomposition of the matrix $I_{a}-P_{0}$ by $(28)$.

$8). (a)$ If the conditions $(25)$ are satisfied, go to $9$; otherwise, Eq $(1)$ has no Re-nnd solutions, and stop. $(b)$ If the condition $(29)$ is satisfied, go to $9$; otherwise, Eq $(1)$ has no Re-pd solutions, and stop.

$9).$ Compute $B$ by $(35)$.

$10).$ Compute the spectral decomposition of the matrices $P_{0}-P_{1}, P_{0}-P_{2}$ by $(36)$ and $(37),$ respectively.

$11).$ Compute $B_{ij}, i, j = 1, 2, 3, 4$ by $(40).$

$12). (a)$ If the conditions $(50)$ are satisfied, go to $13$; otherwise, Eq $(1)$ has no Re-nnd solutions, and stop. $(b)$ If the conditions $(52)$ are satisfied, go to $13$; otherwise, Eq $(1)$ has no Re-pd solutions, and stop.

$13).$ Choosee matrices $M_{12}, S_{12}$ and $M_{34}, S_{34}$, and compute $V_{12}$ and $V_{11}^{(34)}$ by $(9)$ and $(49),$ respectively.

$14). (a)$ Choose matrix $S,$ Hermitian nonnegative definite matrix $T$ and contraction matrix $N$, and compute $H_{11},$ $K$ by $(26)$ and $(41),$ respectively.

$(b)$ Choose matrix $S,$ Hermitian positive definite

matrix $T$ and strict contraction matrix $N$, and compute $H_{11},$ $K$ by $(30)$ and $(44),$ respectively.

$15). (a)$ Choose matrices $V_1, V_3, V_{12}^{(34)}, V_{21}^{(34)}, V_{22}^{(34)}, E$ and $F,$ and compute Re-nnd solutions by $(51).$

$(b)$ Choose matrices $V_1, V_3, V_{12}^{(34)}, V_{21}^{(34)}, V_{22}^{(34)}, H_{12}$ and $R,$ and compute Re-pd solutions by $(53).$

Example 1. Let $m = 8, n = 7, p = 6$, and the matrices $A_{1}, A_{2}, C_{1}, C_{2}$ be given by

It is easy to verify that the conditions $(4), (25)$ and $(50)$ hold. According to Algorithm $1,$ by choosing matrices $M_{12} = 0, S_{12} = 0, M_{34} = 0, S_{34} = 0, S = 0, V_{1} = 0, V_{3} = 0, N = I_{3}$ and $T = \mbox{diag}(1, 1, 0).$ We can obtain a Re-nnd solution $X$ of Eq $(1)$ as follows:

The absolute errors are estimated by

and the eigenvalues of $\frac{1}{2}(X+X^\ast)$ are $(0.0000, 0.1176, 0.3090, 0.7064, 0.9076, 1.3411, 13.8617),$ which implies that $X$ is a Re-nnd matrix.

Example 2. Let $m = 8, n = 7, p = 6$, and the matrices $A_{1}, A_{2}$ be given as in Example $1$, and $C_{1}, C_{2}$ be given by

It is easy to verify that the conditions $(4), (29), (52)$ hold. According to Algorithm $1,$ by choosing matrices $M_{12} = 0, S_{12} = 0, M_{34} = 0, S_{34} = 0, S = 0, V_{1} = 0, V_{3} = 0, T = I_{3}$ and $N = \mbox{diag}(0.5, 0.9).$ We can obtain a Re-pd solution $X$ of Eq $(1)$ as follows:

The absolute errors are estimated by

and the eigenvalues of $\frac{1}{2}(X+X^\ast)$ are $(0.1237, 0.2834, 0.7033, 0.8109, 1.0267, 1.4143, 13.9510),$ which implies that $X$ is a Re-pd matrix.

5.   Conclusions

In the previous sections, we studied the common Re-nonnegative definite and Re-positive definite solutions of linear matrix equations (1). We established a set of necessary and sufficient conditions for the existence of a general common Re-nonnegative definite solution, Re-positive definite solution of (1) respectively. Moreover, we gave the explicit expressions for these general common solutions when the consistent conditions are satisfied. At the end, we showed an algorithm and two examples to illustrate the main results of this paper.