The reverse order law for the weighted least square $g$-inverse of multiple matrix products

1.   Introduction

Let $\mathbb{C}^{m\times n}$ be the set of all $m\times n$ matrices in the complex field. For any $A\in \mathbb{C}^{m\times n}$, we denote the conjugate transpose, the rank, the range space and the null space of $A$ by $A^*$, $r(A)$, $\mathbb{R}(A)$ and $\mathbb{N}(A)$, respectively. In the remainder of this paper, we will adopt

where $A_i\in \mathbb{C}^{m_i\times m_{i+1}}$, $X_i\in \mathbb{C}^{m_{i+1}\times m_i }$ and $X_i$ is called a weighted generalized inverse of $A_i$, $i = 1, 2, \cdots, n$. In particular, $\mathbb{A}_j^{j} = A_j$, $\mathbb{A}_1^{j} = A_1A_2\cdots A_j$, $\mathbb{X}_j^{j} = X_j^*$, $\mathbb{X}_1^{j} = X_1^*X_2^*\cdots X_j^*$, $j = 1, 2, \ldots, n$ and $\mathbb{X}_{n+1}^{n} = I_{m_{n+1}}$.

For $A\in \mathbb{C}^{m\times n}$, let $M\in \mathbb{C}^{m\times m}$, $N\in \mathbb{C}^{n\times n}$ be two positive definite Hermitian matrices. We recall that a weighted generalized inverse $X\in \mathbb{C}^{n\times m}$ of $A$ is a matrix that satisfies some of the following equations ^[1,13]:

We say that $X = A^{(1, 3M)}$ is a $\{1, 3M\}$-inverse or a weighted least squares $g$-inverse of $A$ if $X$ is a common solution of (1) and $(3M)$. Let $A\{1, 3M\}$ denote the set of all $\{1, 3M\}-$inverses of $A$. We say that $X = A^{(1, 4N)}$ is a $\{1, 4N\}$-inverse of $A$ if $X$ is a common solution of (1) and $(4N)$. Let $A\{1, 4N\}$ denote the set of all $\{1, 4N\}-$inverses of $A$. The unique $\{1, 2, 3M, 4N\}$-inverse of $A$ ia called the weighted Moore-Penrose inverse of $A$ and is denoted by $X = A^{(1, 2, 3M, 4N)} = A_{M, N}^{\dagger}$ ^[1,23].

The reverse order law for the weighted generalized inverse of the multiple product of matrices has been widely applied in the theoretic research and numerical computations areas (see ^{[1,2,4,5,8,9,14,23,24]}).

For the very time, Greville ^[6] presented an equivalent condition for the reverse order law $(AB)^{\dagger} = B^{\dagger}A^{\dagger}$. Since then, many authors have studied this problem (see ^{[3,4,7,10,11,15,16,17,19,20,21,22,25,26]}). It is well known that the core problem concerns with the reverse order law and whether conditions

hold, or whether conditions

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

The purpose of this paper is to show some equivalent conditions for the following inclusions

and

where $A_i\in \mathbb{C}^{m_i\times m_{i+1}}$, $i = 1, 2, \ldots, n$ and $M_i\in \mathbb{C}^{m_i\times m_i}$, $i = 1, 2, \ldots, n+1$ are $n+1$ positive definite Hermitian matrices.

2.   Preliminaries

Lemma 2.1. ^[23] Let $L$, $M$ be two complementary subspaces of $\mathbb{C}^m$ and let $P_{L, M}$ be the projector on $L$ along $M$, then

Lemma 2.2. ^[1,23] Let $A\in \mathbb{C}^{m\times n}$, $X\in \mathbb{C}^{n\times m}$ and let $M$ and $N$ be two positive definite Hermitian matrices of order $m$ and $n$, respectively, then

Lemma 2.3. ^[18] Let $A\in \mathbb{C}^{m\times n}$, $B\in \mathbb{C}^{m\times k}$, $C\in \mathbb{C}^{l\times n}$ and $D\in \mathbb{C}^{l\times k}$, and let $M\in \mathbb{C}^{m\times m}$ and $N\in \mathbb{C}^{n\times n}$ be two positive definite Hermitian matrices, then

Lemma 2.4. ^[12] Let $A\in \mathbb{C}^{m\times n}$, $B\in \mathbb{C}^{m\times k}$ and $C\in \mathbb{C}^{p\times n}$, then

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

3.   Main results

Let

be as given in (1.1), and $M_i\in \mathbb{C}^{m_i\times m_i}$, $i = 1, 2, \ldots, n+1$ are positive definite Hermitian matrices. Then, from (2.3) in Lemma 2.2, we know that (1.5) holds if, and only if,

holds for any $X_i\in A_i\{1, 3M_i\}$, $i = 1, 2, \ldots, n$, which is equivalent to:

Hence, we can present the equivalent conditions for (1.5) if the concrete expressions of the maximal rank involved in (3.1) are derived.

Theorem 3.1. Let $A_i\in \mathbb{C}^{m_i\times m_{i+1}}$, $X_i = A_i^{(1, 3M_i)}\in A_i\{1, 3M_i\}$ and $i = 1, 2, \ldots, n$. Let $M_i\in \mathbb{C}^{m_i\times m_i}$, $i = 1, 2, \ldots, n+1$ be positive definite Hermitian matrices and let $\mathbb{A}_i^{j} = A_iA_{i+1}\cdots A_j$, $\enspace 1\leq i\leq j\leq n$ be given as in (1.1). Then,

Proof. From (2.3) in Lemma 2.2 and the definition of the rank of the matrix, we can see that for any $X_i = A_i^{(1, 3M_i)}\in A_i\{1, 3M_i\}$, $i = 1, 2, \ldots, n$, the following three formulas are equivalent:

and

□

Let $\mathbb{X}_i^{j} = X_i^*X_{i+1}^*\cdots X_j^*, \enspace 1\leq i\leq j\leq n$ as in (1.1). Then, from the formula (2.6) in Lemma 2.3 with $A = A_1$, $B = I_{m{_1}}$, $C = {(\mathbb{A}_1^{n})}^*M_1\mathbb{A}_1^{n}{(\mathbb{X}_2^{n})}^*$ and $D = {(\mathbb{A}_1^{n})}^*M_1$, we have

in which by the row or column elementary block operations from the first equality to the second one, we use the rank identities

and

From (3.6) and again by (2.6) in Lemma 2.3 with $A = A_2$, $B = I_{m_{2}}$, $C = {(\mathbb{A}_1^{n})}^*M_1\mathbb{A}_1^{n}{(\mathbb{X}_3^{n})}^*$ and $D = {(\mathbb{A}_1^{n})}^*M_1A_1$, we have

By (2.9) in Lemma 2.4 we have $\left(\begin{matrix}O, &A_2^*\end{matrix}\right)^{\dagger} = \left(\begin{matrix}O\cr (A_2^*)^{\dagger}\end{matrix}\right)$, thus

Combining (3.6) and (3.7) with (3.8), we have

Generally, for $2\leq i\leq n$, we can prove the following fact:

where $\mathbb{A}_1^1 = A_1$ and $\mathbb{X}_{n+1}^{n} = I_{m_{n+1}}$.

In fact, (3.10) is true for $i = 2$ (see (3.9)). Now, assume (3.10) is also true for $i-1{(i\geq3)}$, i.e

Next, we will prove that (3.10) is also true for $i$. By formula (3.11) and (2.6) in Lemma 2.3 with $\tilde{B} = \left(\begin{matrix} I_{m{_i}}, &O, & \cdots, &O\end{matrix}\right)$, $\tilde{A} = A_i$, $\tilde{C} = {(\mathbb{A}_1^{n})}^*M_1\mathbb{A}_1^{n}{(\mathbb{X}_{i+1}^{n})}^*$, $\tilde{D} = \left(\begin{matrix}{(\mathbb{A}_1^{n})}^*M_1\mathbb{A}_1^{i-1}, & -{(\mathbb{A}_1^{n})}^*M_1\mathbb{A}_1^{i-2}M_{i-1}^{-1} F_{A_{i-1}^*}, & \cdots, & -{(\mathbb{A}_1^{n})}^*M_1A_1M_2^{-1} F_{A_2^*}\end{matrix}\right)$ and $\tilde{E} = {(\mathbb{A}_1^{n})}^*M_1\mathbb{A}_1^{n}{(\mathbb{X}_{i}^{n})}^*-{(\mathbb{A}_1^{n})}^*M_1\mathbb{A}_1^{i-1}$, we have

By the row or column elementary block operations of formula (2.9) in Lemma 2.4 we have,

and

where

and

In particular, when $i = n$, we get $\mathbb{A}_1^1 = A_1$, $\mathbb{X}_{n+1}^{n} = I_{m_{n+1}}$ and

Applying (3.13) with Lemma 2.4, we finally have

According to the formulas (1.1), (3.3)–(3.5) and (3.14), we have

From Lemmas 2.1, 2.4 and Theorem 3.1, we have:

Corollary 3.1. Let $A_i\in \mathbb{C}^{m_i\times m_{i+1}}$, $X_i = A_i^{(1, 3M_i)}\in A_i\{1, 3M_i\}$. Let $M_i\in \mathbb{C}^{m_i\times m_i}$ be positive definite Hermitian matrices $i = 1, 2, \ldots, n+1$, and let $\mathbb{A}_i^{j} = A_iA_{i+1}\cdots A_j, \; 1\leq i\leq j\leq n$ be given as in (1.1). Then, the following statements are equivalent:

(1) $A_n\{1, 3M_n\}A_{n-1}\{1, 3M_{n-1}\}\cdots A_1\{1, 3M_1\}\subseteq (A_1A_2\cdots A_n)\{1, 3M_1\}$;

(2) $r\left(\begin{matrix}\ A_n^*M_n & O & \cdots & O \cr O & A_{n-1}^*M_{n-1} & \cdots &O \cr \vdots & \vdots & \ddots & \vdots \cr O &O & \cdots &A_2^*M_2\cr {(\mathbb{A}_1^{n})}^*M_1\mathbb{A}_1^{n-1} & {(\mathbb{A}_1^{n})}^*M_1\mathbb{A}_1^{n-2} & \cdots & {(\mathbb{A}_1^{n})}^*M_1\mathbb{A}_1^{1}\end{matrix}\right) = \sum\limits_{i = 2}^n r(A_i)$;

(3) $\mathbb{R}{({(\mathbb{A}_1^{i-1})}^*M_1\mathbb{A}_1^{n})}\subseteq \mathbb{R}{(M_iA_i)}$, $i = 2, 3, \ldots, n$;

(4) $A_{i}(A_i)_{M_i, I_{m_{i+1}}}^{\dagger} M_i^{-1}{(\mathbb{A}_1^{i-1})}^*M_1\mathbb{A}_1^{n} = M_i^{-1}{(\mathbb{A}_1^{i-1})}^*M_1\mathbb{A}_1^{n}$, $i = 2, 3, \ldots, n$.

Proof. According to Theorem 3.1, we get that (1) and (2) are equivalent since

Next, we will prove $(3)\Longleftrightarrow(2)$. From (3.16) and (2.8) in Lemma 2.4, we have

According to (3.17), we have $r(M_iA_i) = r(A_i)$ and $(3)\Longleftrightarrow(2)$ if, and only if,

From Lemmas 2.1 and 2.4, we have $E_{M_iA_i} = I_{m_i}-(M_iA_i)(M_iA_i)^{\dagger} = I_{m_i}-P_{\mathbb{R}(M_iA_i), \mathbb{N}(A_i^*M_i)}$ and $(3)\Longleftrightarrow(2)$, where $i = 2, 3, \ldots, n$.

By using formula (3) and Lemma 2.1, we get $(3)\Longleftrightarrow(4)$ since

where $i = 2, 3, \ldots, n$. $\square$ □

From Lemma 2.2, we have $X\in A\{1, 4N\}\Leftrightarrow X^*\in A^*\{1, 3N^{-1}\}$, so from Theorem 3.1 and Corollary 3.1, we have

Theorem 3.2. Let $A_i\in \mathbb{C}^{m_i\times m_{i+1}}$, $X_i = A_i^{(1, 4N_{i+1})}\in A_i\{1, 4N_{i+1}\}$. Let $N_i\in \mathbb{C}^{m_i\times m_i}$ be positive definite Hermitian matrices $i = 1, 2, \ldots, n+1$, and let $\mathbb{A}_i^{j} = A_iA_{i+1}\cdots A_j, \; 1\leq i\leq j\leq n$ be given as in (1.1). Then,

From Lemmas 2.1, 2.4 and Theorem 3.2, we have

Corollary 3.2. Let $A_i\in \mathbb{C}^{m_i\times m_{i+1}}$, $X_i = A_i^{(1, 4N_{i+1})}\in A_i\{1, 4N_{i+1}\}$. Let $N_i\in \mathbb{C}^{m_i\times m_i}$ be positive definite Hermitian matrices $i = 1, 2, \ldots, n+1$, and let $\mathbb{A}_i^{j} = A_iA_{i+1}\cdots A_j, \; 1\leq i\leq j\leq n$ be given as in (1.1). Then, the following statements are equivalent:

(1) $A_n\{1, 4N_{n+1}\}A_{n-1}\{1, 4N_{n}\}\cdots A_1\{1, 4N_2\}\subseteq (A_1A_2\cdots A_n)\{1, 4N_{n+1}\}$;

(2) $r\left(\begin{matrix}\ N_2^{-1} A_1^* & O & \cdots & O & \mathbb{A}_2^{n}N_{n+1}^{-1}{(\mathbb{A}_1^{n})}^* \cr O &N_3^{-1} A_{2}^* & \cdots &O & \mathbb{A}_3^{n}N_{n+1}^{-1}{(\mathbb{A}_1^{n})}^* \cr \vdots & \vdots & \ddots & \vdots & \vdots \cr O & O & \cdots &N_{n}^{-1}A_{n-1}^* & \mathbb{A}_n^{n}N_{n+1}^{-1}{(\mathbb{A}_1^{n})}^*\end{matrix}\right) = \sum\limits_{i = 1}^{n-1} r(A_i)$;

(3) $\mathbb{R}{(\mathbb{A}_{i+1}^{n}N_{n+1}^{-1}{(\mathbb{A}_{1}^{n})}^*)}\subseteq \mathbb{R}{(N_{i+1}^{-1}A_i^*)}$, $i = 1, 2, \ldots, {n-1}$;

(4) $(A_i)_{I_{m_{i}, N_{i+1}}}^{\dagger}A_{i} {\mathbb{A}_{i+1}^{n}N_{n+1}^{-1}{(\mathbb{A}_{1}^{n})}^*} = {\mathbb{A}_{i+1}^{n}N_{n+1}^{-1}{(\mathbb{A}_{1}^{n})}^*}$, $i = 1, 2, \ldots, {n-1}$.

4.   Conclusions

The reverse order law for the weighted generalized inverses of the multiple product of matrices has been studied in this article by using the ranks of the generalized Schur complement. The work in this paper was a useful tool in many algorithms for the computation of the weighted least squares technique of matrix equations.

Use of AI tools declaration

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

Acknowledgments

The authors wish to thank Professor Jie Gong and the referees. This work was supported by the project for characteristic innovation of 2018 Guangdong University (No: 2018KTSCX234), the National Natural Science Foundation of China (No: 11771159), 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.