The reverse order laws for $\{1, 2, 3M\}$- and $\{1, 2, 4N\}$- inverse of three matrix products

1.   Introduction

At first, we will provide the following explanations of some of the notations for the convenience of readers.

Definition 1.1. $C^{m\times n}$: the set of $m\times n$ complex matrices,

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; I_k$: $k$-order identity matrix,

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; O_{m\times n}$: zero matrix of order $m\times n$,

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; R(A)$, $N(A)$: the range space and the null space of $A$, respectively,

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; r(A)$, $A^*$: the rank and the conjugate transpose of $A$, respectively.

Definition 1.2. ^[1,2] Let $A\in C^{m\times n}$, and let $M\in C^{m\times m}$, $N\in C^{n\times n}$ be two positive definite Hermitian matrices. $X$ is the weighted Moore-Penrose inverse of $A$ when it satisfies

where $X$ is denoted by $X = A^{(1, 2, 3M, 4N)} = A_{M, N}^{\dagger}$.

For a given matrix $A\in C^{m\times n}$, the sets $A\{1, 2, 3M\}$- and $A\{1, 2, 4N\}-$ inverses of $A$ are

$A\{1, 2, 3M\} = \left\{X\in C^{n\times m}\; |\; AXA = A, \; XAX = X, \; (MAX)^* = MAX\right\},$

$A\{1, 2, 4N\} = \left\{X\in C^{n\times m}\; |\; AXA = A, \; XAX = X, \; (NXA)^* = NXA\right\};$

more relevant theories can be found in ^[1,3].

The reverse order law for weighted generalized inverses is a key tool in the study of the weighted least squares problem, the weighted perturbation theory, optimization problems, and other related topics ^[4,5,6].

The reverse order laws for generalized inverses of matrix products are a class of interesting problems that are fundamental in the theory of generalized inverses ^[7,8,9,10]. In 1966, Greville ^[7] first gave an equivalent condition for the so-called reverse order law $B^{\dagger}A^{\dagger} = (AB)^{\dagger}$. Since then, many authors have studied this problem ^{[11,12,13,14,15]}. On studying the reverse order for any $\{i, j, \cdots, k\}$-inverse of matrix products, one important relations problem is: under what conditions

holds, where $\{(i), (j), \cdots, (k)\}\subseteq \{(1), (2), (3M), (4N)\}$.

In ^[16], some necessary and sufficient conditions were presented for the first times for several types of reverse order laws to hold for weighted generalized inverses. Since then, reverse order laws for weighted generalized inverses of matrix products have attracted considerable attention and some interesting results have been derived ^{[17,18,19,20]}.

The purpose of this paper is to show some equivalent conditions for the following so-called reverse order laws

and

where $A_i\in C^{m_i\times m_{i+1}}, i = 1, 2, 3$, $M_i\in C^{m_i\times m_i}, i = 1, 2, 3$, $N_i\in C^{m_i\times m_i}, and\; \; i = 2, 3, 4$ are six positive definite Hermitian matrices.

The following lemmas are essential to the rest of this paper.

Lemma 1.1. ^[3] Let $L$, $M$ be two complementary subspaces of $C^m$, and let $P_{L, M}$ be the projector on $L$ along $M$, then

Lemma 1.2. ^[1,3] Let $A\in C^{m\times n}$, $X\in C^{n\times m}$, and let $M$, $N$ be two positive definite Hermitian matrices of order $m$ and $n$, respectively, then

Lemma 1.3. ^[21] Let $A\in C^{m\times n}$, $B\in C^{m\times k}$, $C\in C^{l\times n}$, $D\in C^{l\times k}$, and let $M\in C^{m\times m}$, $N\in C^{n\times n}$ be two positive definite Hermitian matrices, then

Lemma 1.4. ^[22] Let $A\in C^{m\times n}$, $B\in C^{m\times k}$ and $C\in C^{p\times n}$, then

where the projectors $E_A = I_m-AA^{\dagger}$, $E_B = I_m-BB^{\dagger}$, $F_A = I_n-A^{\dagger}A$, $F_C = I_n-C^{\dagger}C$.

2.   Main results

In this section, we will present some necessary and sufficient conditions for the reverse order laws for the weighted generalized inverses $\{1, 2, 3M\}-$ and $\{1, 2, 4N\}-$ of three matrix products. The following theorem is the main result in this section.

Theorem 2.1. Let $A_i\in C^{m_i\times m_{i+1}}$, $A_i^{(1, 2, 3M_i)}\in A_i\{1, 2, 3M_i\}$ and $i\in \{1, 2, 3\}$. Let $M_i\in C^{m_i\times m_i}$, $i\in \{1, 2, 3\}$ be three positive definite Hermitian matrices, then

Proof. According to the formula $(1.4)$ in Lemma $1.2$, for any $A_i^{(1, 2, 3M_i)}\in A_i\{1, 2, 3M_i\}$, $i\in \{1, 2, 3\}$, we can reach the conclusion that

holds if and only if

holds, which are respectively equivalent to the following two identities

and

Using the formula $(1.7)$ in Lemma $1.3$ with $A = A_1$, $B = I_{m{_1}}$, $D = A_3^*A_2^*A_1^*M_1$ and $C = A_3^*A_2^*A_1^*M_1A_1A_2A_3A_3^{(1, 2, 3M_3)}A_2^{(1, 2, 3M_2)}$, we have

From (2.4) and again by formula $(1.7)$ in Lemma $1.3$ with $A = A_2$, $B = I_{m{_2}}$, $D = A_3^*A_2^*A_1^*M_1A_1$, and $C = A_3^*A_2^*A_1^*M_1A_1A_2A_3A_3^{(1, 2, 3M_3)}$, we have

According to the formulas $(1.12)$ and $(1.13)$ of Lemma $1.4$, we have

and

and

Combining (2.5), (2.6), and (2.7), we have

From (2.8) and again by formula $(1.7)$ in Lemma $1.3$ with $A = A_3$, $B = (I_{m{_3}}, O)$, $C = A_3^*A_2^*A_1^*M_1A_1A_2A_3$, and $D = (A_3^*A_2^*A_1^*M_1A_1A_2, \; \; A_3^*A_2^*A_1^*M_1A_1M_2^{-1}F_{A_2^*})$, we have

and

According to $(2.8)$ and formula $(1.12)$ of the Lemma $1.4$, we have

Combining $(2.2)$ and (2.10), we have

if and only if

In the rest of the section, we will find the equivalent conditions of $(2.3)$. By Lemma $1.3 \; (1.7)$ with $A = A_1$, $B = I_{m_{1}}$, $C = A_3^{(1, 2, 3M_3)}A_2^{(1, 2, 3M_2)}$, and $D = O$, we have

Form (2.12) and using Lemma $1.3 \; (1.7)$ with $A = A_2$, $B = I_{m_{2}}$, $C = A_3^{(1, 2, 3M_3)}$, and $D = O$, we have

From (2.13) and formula $(1.7)$ of Lemma $1.3$, we have

Combining (2.13) and (2.14), we have

Since $r(A_3^{(1, 2, 3M_3)}) = r(A_3)$, then

On the other hand, according to $(1.8)$ of Lemma $1.3$ with $A = A_3$, $B = A_2^{(1, 2, 3M_2)}A_1^{(1, 2, 3M_1)}$, $C = I_{m_{4}}$, and $D = O$, we have

From (2.17) and again by formula $(1.8)$ in Lemma $1.3$ with $A = A_2$, $B = A_1^{(1, 2, 3M_1)}$, $C = A_3^*M_3$, and $D = O$, we have

By formula $(1.12)$ of Lemma $1.4$, we have

and

Combining Eqs (2.19) and (2.20), we obtain Eq (2.18).

Using Eq (2.18) and formula $(1.8)$ from Lemma $1.3$ with $(A = A_1)$, $(B = (O, \; -I_{m_1}))$, $(C = A_{2}^*{M_2})$, and $(D = (A_{2}^*{M_2}A_{2}M_{3}^{-1}F_{A_3^*}, \; O))$, we derive

where

and

By the formula (2.23), we have

By the formula (1.12) of Lemma $1.4$, we have

Combining (2.21), (2.22), (2.24), and (2.25), we have

According to (2.2), (2.3), (2.11), (2.16), and (2.26), we have proved Theorem 2.1. $\; \; \; \; \; \; \; \; \; \; \square$

From Lemma $1.1$, Lemma $1.4$, and Theorem $2.1$, we immediately obtain the following corollary.

Corollary 2.1. Let $A_i\in C^{m_i\times m_{i+1}}$, $A_i^{(1, 2, 3M_i)}\in A_i\{1, 2, 3M_i\}$, where $i\in \{1, 2, 3\}$. Let $M_i\in C^{m_i\times m_i}$, $i\in \{1, 2, 3\}$ be three positive definite Hermitian matrices. Then the following statements are equivalent:

(1)

(2)

and

(3)

and

(4)

and

Proof. According to Theorem $2.1$, we have $(1)\Leftrightarrow (2)$. Now, we will prove $(3)\Rightarrow (2)$. By $(2.30)$ and $(2.31)$, we have $R{(M_3^{-1}A_2^*A_1^*M_1A_1A_2A_3)}\subseteq R{(A_3)}$ and $R{(M_2^{-1}A_1^*M_1A_1A_2A_3)}\subseteq R{(A_2)}$. Using Lemma 1.1, we have

Next, we will prove $(2)\Rightarrow (3)$, that is, $(2.28) \; \Rightarrow \; (2.30)$ and $(2.31)$. According to $(2.28)$ and the formula $(1.11)$ of Lemma $1.4$, we have

and

and

According to (1.2) of Lemma $1.1$, we have $(2) \; \Rightarrow$ $(3)$.

Finally, we will prove $(3) \; \Longleftrightarrow \; (4)$, that is $(2.30)\Leftrightarrow (2.32)$ and $(2.31)\Leftrightarrow (2.33)$. According to $(1.2)$ of Lemma $1.1$, we have

and

and

So we have that $(1)$, $(2)$, $(3)$, and $(4)$ are equivalent.□

From Lemma $1.2$, we have $X\in A\{1, 2, 4N\}$ if and only if $X^*\in A^*\{1, 2, 3N^{-1}\}$. Thus according to the results obtained in Theorem $2.1$ and Corollary $2.1$, we obtain the following theorem and corollary without any proof.

Theorem 2.2. Let $A_i\in C^{m_i\times m_{i+1}}$, $A_i^{(1, 2, 4N_{i+1})}\in A_i\{1, 2, 4N_{i+1}\}$, where $i\in \{1, 2, 3\}$. Let $N_i\in C^{m_i\times m_i}$, $i\in \{1, 2, 3, 4\}$ be four positive definite Hermitian matrices. Then

From Lemma $1.1$, Lemma $1.4$, and Theorem $2.2$, we immediately obtain the corollary as follows:

Corollary 2.2. Let $A_i\in C^{m_i\times m_{i+1}}$, $A_i^{(1, 2, 4N_{i+1})}\in A_i\{1, 2, 4N_{i+1}\}$, where $i\in \{1, 2, 3\}$. Let $N_i\in C^{m_i\times m_i}$, $i\in \{1, 2, 3, 4\}$ be four positive definite Hermitian matrices. Then the following statements are equivalent:

(1)

(2)

and

(3)

and

(4)

and

3.   Conclusions

The reverse order law for the inverses $\{1, 2, 3M\}$- and $\{1, 2, 4N\}$- of matrix products has been studied in this article by using the ranks of the generalized Schur complement. The work performed in this paper is a useful tool for many algorithms for the computation of the weighted least squares technique of matrix equations.

Author contributions

Baifeng Qiu: Resources; Yingying Qin: Resources; Zhiping Xiong: Conceptualization, writing-review and editing. All authors have read and agree to the published version of the manuscript..

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to thank the editors and the anonymous referees for their very detailed comments and constructive suggestions, which greatly improved the presentation of this paper. This work was supported by the project for characteristic innovation of 2018 Guangdong University (No: 2018KTSCX234) and the joint research and Development fund of Wuyi University, Hong Kong and Macao (No: 2019WGALH20) and the basic Theory and Scientific Research of Science and Technology Project of Jiangmen City, China (No: 2021030102610005049).

Conflict of interest

The authors declare that there are no conflicts of interest.